**2. Methods**

As shown in Figure 1, the proposed method of discrepancy analysis comprises two main parts: (i) classifying hyperspectral UAV images using the hybrid CNN for generating the land cover map and (ii) comparing the consistency between the cadastral map and land cover map for detecting inconsistent parcels through a query-based approach.

**Figure 1.** Proposed process of discrepancy analysis. UAV: unmanned aerial vehicle.


Step 3. The extended mapping G is a left inverse of the Fourier transform F. For this, observe that

$$(f,g) = \int\_{\mathbb{R}} (\mathcal{F}f)(x) \overline{(\mathcal{F}g)(x)} \, d\rho(x)$$

for all f,g ∈ L2 r(a, b), since F is isometric by Lemma B.1.1. Thus, by means of the Fubini theorem one concludes that for f,g ∈ L2 r(a, b) and g having compact support,

$$\begin{aligned} (f,g) &= \lim\_{\eta \to \mathbb{R}} \int\_{\eta} (\mathcal{F}f)(x) \left( \int\_{a}^{b} \omega(t,x) \overline{g(t)} r(t) dt \right) d\rho(x) \\ &= \lim\_{\eta \to \mathbb{R}} \int\_{a}^{b} \left( \int\_{\eta} \omega(t,x) (\mathcal{F}f)(x) \, d\rho(x) \right) \overline{g(t)} r(t) dt \\ &= (\mathcal{G}\mathcal{F}f,g), \end{aligned}$$

where, in the last step, (B.1.10) and the continuity of the inner product have been used. Since the functions with compact support are dense in L2 r(a, b), one obtains GFf = f for all f ∈ L2 r(a, b). -

Recall that multiplication by the independent variable in the Hilbert space L2 dρ(R) generates a self-adjoint operator Q whose resolvent is given by

$$(Q - \lambda)^{-1} = \frac{1}{x - \lambda}, \qquad \lambda \in \mathbb{C} \backslash \mathbb{R}.$$

Hence, by (B.1.7), one sees that

$$\left( \left( A - \lambda \right)^{-1} f, g \right) = \left( \left( Q - \lambda \right)^{-1} \mathcal{F} f, \mathcal{F} g \right)\_{\rho} = \left( \mathcal{F}^\* \left( Q - \lambda \right)^{-1} \mathcal{F} f, g \right)\_{\rho}$$

for all f,g ∈ L2 r(a, b), which leads to

$$(A - \lambda)^{-1} = \mathcal{F}^\*(Q - \lambda)^{-1}\mathcal{F}, \qquad \lambda \in \mathbb{C} \backslash \mathbb{R}. \tag{B.1.11}$$

In general, here the Fourier transform F is an isometry, which is not necessarily onto.

The above results have been proved under the assumption that ω is a continuous function on (a, b) × R which satisfies (B.1.2). For the following theorem one needs the additional condition:

$$\begin{array}{c} \text{for each } x\_0 \in \mathbb{R} \text{ there exists a compactly} \\ \text{supported function } f \in L\_r^2(a, b) \text{ such that } (\mathcal{F}f)(x\_0) \neq 0. \end{array} \tag{B.1.12}$$

In the present situation this condition is equivalent to the following:

for each x0 ∈ (a, b) there exists t0 ∈ R such that ω(t0, x0) = 0. (B.1.13)

**Theorem B.1.4.** Let ω be continuous on (a, b)×R and assume that (B.1.2) and one of the equivalent conditions (B.1.12) or (B.1.13) are satisfied. Then the Fourier transform

$$f \mapsto \widehat{f}, \quad \widehat{f}(x) = \int\_{a}^{b} \omega(t, x) f(t) \, r(t) \, dt, \quad x \in \mathbb{R},$$

extends by continuity from the compactly supported functions f ∈ L2 r(a, b) to a unitary mapping F : L2 r(a, b) → L2 dρ(R). Moreover, the self-adjoint operator A in L2 r(a, b) is unitarily equivalent to the multiplication operator Q by the independent variable in L2 dρ(R) via the Fourier transform F:

$$A = \mathcal{F}^\* Q \mathcal{F}. \tag{B.1.14}$$

Proof. It is clear from Lemma B.1.1 that F : L2 r(a, b) → L2 dρ(R) is isometric and hence it remains to show for the first part of the theorem that F is surjective.

Assume that ψ ∈ L2 dρ(R), ψ = 0, is orthogonal to ran F. Note that

$$(\psi, \mathcal{F}f)\_\rho = 0 \quad \Rightarrow \quad (\overline{\psi}, \mathcal{F}\overline{f})\_\rho = 0$$

and thus it follows that Re ψ and Im ψ are also orthogonal to ran F; hence it is no restriction to assume that ψ ∈ L2 dρ(R) is real. Since ψ = 0, there exist an interval I = [α, β] and a Borel set B ⊂ I with positive ρ-measure, such that ψ(x) > 0 (or ψ(x) < 0) for all x ∈ B. By the condition (B.1.12), there exists for each x0 ∈ I a compactly supported function fx0 ∈ L2 r(a, b) such that (Ffx0 )(x0) > 0. Since Ffx0 is continuous, there exists an open interval Ix0 containing x0 such that (Ffx0 )(x) > 0 for all x ∈ Ix0 . As I is compact there exist finitely many points x1,...,xn ∈ I such that

$$I \subset \bigcup\_{i=1}^{n} I\_{x\_i}.$$

Hence, B ∩ Ixj has positive ρ-measure for some j ∈ {1,...,n} and therefore

$$(\chi\_{B \cap I\_{x\_j}} \psi, \mathcal{F}f\_{x\_j})\_\rho = \int\_{\mathbb{R}} \chi\_{B \cap I\_{x\_j}}(x) \psi(x) (\mathcal{F}f\_{x\_j})(x) \, d\rho(x) \neq 0. \tag{B.1.15}$$

On the other hand, since ψ ⊥ ran F, Corollary B.1.2 implies that

$$(\chi\_{\delta}\psi, \mathcal{F}f)\_{\rho} = (\psi, \chi\_{\delta}\mathcal{F}f)\_{\rho} = (\psi, \mathcal{F}E(\delta)f)\_{\rho} = 0$$

for all f ∈ L2 r(a, b) and all bounded open intervals δ ⊂ R. Hence, by the regularity of the Borel measure ρ, also (χB∩Ixj ψ, Ff)ρ = 0 for all f ∈ L2 r(a, b); this contradicts (B.1.15). Therefore, ran F is dense in L2 dρ(R), and since F : L2 r(a, b) → L2 dρ(R) is isometric, one obtains that F is surjective.

The identity (B.1.14) follows from (B.1.11), the fact that F is unitary, and Lemma 1.3.8. -

In the situation of Theorem B.1.4 the inverse of the Fourier transform F is actually given by the reverse Fourier transform G in Lemma B.1.3, which is now a unitary mapping from L2 dρ(R) to L2 r(a, b). Thus, for all ϕ ∈ L2 dρ(R) one has

$$\lim\_{\eta \to \mathbb{R}} \int\_{a}^{b} \left| \mathcal{F}^{-1} \varphi(t) - \int\_{\eta} \omega(t, x) \varphi(x) \, d\rho(x) \right|^2 r(t) dt = 0.$$

## **B.2 The vector case**

Let (a, b) be an open interval and let Δ be a measurable nonnegative 2 × 2 matrix function on (a, b). Let A be a self-adjoint relation in the corresponding Hilbert space L2 Δ(a, b) and assume that its multivalued part is at most finite-dimensional. Let E(·) be the spectral measure of Aop, where Aop is the orthogonal operator part of A: it is a self-adjoint operator in L2 Δ(a, b) mul A. In this case the basic ingredient is a continuous 1×2 matrix function (t, x) → ω(t, x) on (a, b)×R whose entries are real. Hence, the Fourier transform f of a compactly supported function f ∈ L2 Δ(a, b) given by

$$
\widehat{f}(x) = \int\_a^b \omega(t, x)\Delta(t)f(t)\,dt, \quad x \in \mathbb{R}, \tag{B.2.1}
$$

is a well-defined complex function that is continuous on R. The main assumption is that there exists a nondecreasing function ρ on R such the identity

$$\widehat{f}(E(\delta)f,g) = \int\_{\delta} \widehat{f}(x)\overline{\widehat{g}(x)}\,d\rho(x), \quad \delta \subset \mathbb{R},\tag{B.2.2}$$

holds for all f,g ∈ L2 Δ(a, b) with compact support and all bounded open intervals δ ⊂ R, whose endpoints are not eigenvalues of Aop. An approximation argument shows that (B.2.2) remains valid for all f,g ∈ L2 Δ(a, b) with compact support and all bounded open intervals δ ⊂ R.

It follows from the assumption (B.2.2) that for every function f ∈ L2 Δ(a, b) with compact support the continuous function f belongs to L2 dρ(R). In the present situation Lemma B.1.1 remains valid in a slightly modified form; here F is a partial isometry with ker F = mul A.

**Lemma B.2.1.** The Fourier transform f → f in (B.2.1) extends by continuity from the compactly supported functions in L2 Δ(a, b) to a partial isometry

$$\mathcal{F}: L^2\_{\Delta}(a, b) \to L^2\_{d\rho}(\mathbb{R})$$

with ker F = mul A such that for all f ∈ L2 Δ(a, b)

$$\lim\_{\alpha \to a, \beta \to b} \int\_{\mathbb{R}} \left| (\mathcal{F}f)(x) - \int\_{\alpha}^{\beta} \omega(t, x) \Delta(t) f(t) \, dt \right|^2 \, d\rho(x) = 0.$$

The Fourier transform F and the spectral measure E(·) are related via

$$\mathbb{E}\left(E(\delta)f,g\right) = \int\_{\delta} (\mathcal{F}f)(x) \overline{(\mathcal{F}g)(x)} \, d\rho(x) \tag{B.2.3}$$

for all f,g ∈ L2 Δ(a, b), where δ ⊂ R is any bounded open interval. Next one verifies as in the proof of Corollary B.1.2 that the identity

$$
\mathcal{F}(E(\delta)f) = \chi\_{\delta} \mathcal{F}f, \quad f \in L^2\_{\Delta}(a,b),
$$

holds for bounded open intervals δ ⊂ R. Further, (B.2.3) and an approximation argument using characteristic functions show that

$$\left( (A - \lambda)^{-1} f, g \right) = \int\_{\mathbb{R}} \frac{\mathcal{F}f(x) \overline{\mathcal{F}g(x)}}{x - \lambda} d\rho(x) \tag{B.2.4}$$

for all f,g ∈ L2 Δ(a, b) and λ ∈ ρ(A). Moreover,

$$\left( (A - \lambda)^{-1} f \right)(t) = \int\_{\mathbb{R}} \frac{\omega(t, x)^{\*}}{x - \lambda} \, \mathcal{F}f(x) \, d\rho(x), \quad \lambda \in \mathbb{C} \,\backslash \,\mathbb{R},\tag{B.2.5}$$

for almost all t ∈ (a, b) and, in particular, the integrand on the right-hand side is integrable for almost all t ∈ (a, b).

The reverse Fourier transform of a function ϕ ∈ L2 dρ(R) with compact support is defined by

$$
\varphi(t) = \int\_{\mathbb{R}} \omega(t, x)^\* \varphi(x) \, d\rho(x), \quad t \in (a, b). \tag{B.2.6}
$$

Then ˘ϕ is a well-defined 2 × 1 matrix function which is continuous on (a, b). By means of Lemma B.2.1 one now obtains the following result, which is similar to Lemma B.1.3. There are some slight differences in the proof for the vector case, which is provided here for completeness.

**Lemma B.2.2.** The reverse Fourier transform ϕ → ϕ˘ in (B.2.6) extends by continuity from the compactly supported functions in L2 dρ(R) to a contractive mapping G : L2 dρ(R) → L2 Δ(a, b) such that for all ϕ ∈ L2 dρ(R)

$$\lim\_{\eta \to \mathbb{R}} \int\_{a}^{b} \left| \mathcal{G}\varphi(t) - \int\_{\eta} \omega(t, x)^{\*} \varphi(x) \, d\rho(x) \right|^{2} dt = 0. \tag{B.2.7}$$

In fact, the extension G satisfies GFf = f for all f ∈ L2 Δ(a, b) mul A.

Proof. Step 1. The mapping ϕ → ϕ˘ takes the compactly supported functions in L2 dρ(R) contractively into L2 Δ(a, b). For this, let ϕ ∈ L2 dρ(R) have compact support and let [α, β] ⊂ (a, b) be a compact interval. First observe that χ[α,β]ϕ˘ ∈ L2 Δ(a, b), so that indeed

$$\int\_{\alpha}^{\beta} \omega(t, x) \Delta(t) \vec{\varphi}(t) \, dt = (\mathcal{F}(\chi\_{[\alpha, \beta]} \vec{\varphi}))(x).$$

Therefore, one obtains

$$\begin{split} \int\_{\alpha}^{\beta} \psi(t)^{\*} \Delta(t) \psi(t) \, dt &= \int\_{\alpha}^{\beta} \left( \int\_{\mathbb{R}} \omega(t,x)^{\*} \varphi(x) \, d\rho(x) \right)^{\*} \Delta(t) \psi(t) \, dt \\ &= \int\_{\mathbb{R}} \left( \int\_{\alpha}^{\beta} \omega(t,x) \Delta(t) \varphi(t) \, dt \right) \overline{\varphi(x)} \, d\rho(x) \\ &= \int\_{\mathbb{R}} (\mathcal{F}(\chi\_{[\alpha,\beta]} \varphi))(x) \overline{\varphi(x)} \, d\rho(x) \\ &\leq \|\mathcal{F}(\chi\_{[\alpha,\beta]} \varphi)\|\_{\rho} \|\varphi\|\_{\rho} \leq \|\chi\_{[\alpha,\beta]} \psi\| \|\varphi\|\_{\rho}, \end{split}$$

where Lemma B.2.1 was used in the last step. The above estimate gives for any compact interval [α, β] ⊂ (a, b)

$$\sqrt{\int\_{\alpha}^{\beta} \phi(t)^{\*} \Delta(t) \phi(t) \, dt} \le \|\varphi\|\_{\rho}.$$

By the monotone convergence theorem this leads to the inequality

 ϕ˘ ≤ ϕ ρ.

Hence, the mapping ϕ → ϕ˘ takes the compactly supported functions in L2 dρ(R) contractively into L2 Δ(a, b).

Step 2. The mapping ϕ → ϕ˘ defined on the functions in L2 dρ(R) that have compact support can be contractively extended to all of L2 dρ(R). To see this, let ϕ ∈ L2 dρ(R) and approximate ϕ in L2 dρ(R) by functions ϕn ∈ L2 dρ(R) with compact support. Then ϕn − ϕm is a Cauchy sequence in L2 dρ(R) and since

$$\|\not\varphi\_n - \not\varphi\_m\| \le \|\varphi\_n - \varphi\_m\|\_\rho,$$

there is an element f ∈ L2 Δ(a, b) such that ˘ϕn → f in L2 Δ(a, b). It follows from ϕ˘n ≤ ϕn ρ that in fact

$$\|f\| \le \|\varphi\|\_{\rho^\*}$$

In particular, the mapping ϕ → f from L2 dρ(R) to L2 Δ(a, b) is a contraction. Hence, the operator G given by Gϕ = f is well defined and takes L2 dρ(R) contractively to L2 Δ(a, b). The assertion in (B.2.7) is clear by multiplying ϕ ∈ L2 dρ(R) by appropriately chosen characteristic functions.

Step 3. The extended mapping G is a left inverse of the Fourier transform F. For this, first note that it follows from (B.2.3) and dominated convergence that

$$(f,g) = \int\_{\mathbb{R}} (\mathcal{F}f)(x) \overline{(\mathcal{F}g)(x)} \, d\rho(x)$$

for all f ∈ L2 Δ(a, b) mul A and g ∈ L2 Δ(a, b). Thus, by means of the Fubini theorem, one concludes for f ∈ L2 Δ(a, b) mul A and g ∈ L2 Δ(a, b), each having compact support, that

$$\begin{aligned} \mathcal{I}(f,g) &= \lim\_{\eta \to \mathbb{R}} \int\_{\eta} (\mathcal{F}f)(x) \left( \int\_{a}^{b} g(t)^{\*} \Delta(t) \omega(t,x)^{\*} \, dt \right) \, d\rho(x) \\ &= \lim\_{\eta \to \mathbb{R}} \int\_{a}^{b} g(t)^{\*} \Delta(t) \left( \int\_{\eta} \omega(t,x)^{\*} (\mathcal{F}f)(x) \, d\rho(x) \right) \, dt \\ &= (\mathcal{G}\mathcal{F}f,g), \end{aligned}$$

where, in the last step, (B.2.7) and the continuity of the inner product have been used. This implies

$$\mathcal{G}\mathcal{F}f = f$$

for all f ∈ L2 Δ(a, b) mul A with compact support. Since the functions in L2 Δ(a, b) with compact support are dense in L2 Δ(a, b) and mul A is finite-dimensional by assumption, it follows that the functions with compact support are also dense in L2 Δ(a, b) mul A. Therefore, one obtains from the contractivity of F and G that GFf = f for all f ∈ L2 Δ(a, b) mul A. -

Recall that multiplication by the independent variable in the Hilbert space L2 dρ(R) generates a self-adjoint operator Q whose resolvent is given by

$$(Q - \lambda)^{-1} = \frac{1}{x - \lambda}, \qquad \lambda \in \mathbb{C} \backslash \mathbb{R}.$$

Hence, by (B.2.4), one sees that

$$\left( (A - \lambda)^{-1} f, g \right) = \left( (Q - \lambda)^{-1} \mathcal{F} f, \mathcal{F} g \right)\_{\rho} = \left( \mathcal{F}^\* (Q - \lambda)^{-1} \mathcal{F} f, g \right)\_{\rho}$$

for all f ∈ L2 Δ(a, b) and g ∈ L2 Δ(a, b), which leads to

$$(A - \lambda)^{-1} = \mathcal{F}^\*(Q - \lambda)^{-1}\mathcal{F}, \qquad \lambda \in \mathbb{C} \backslash \mathbb{R}. \tag{B.2.8}$$

Consider the restriction Fop : L2 Δ(a, b) mul A → L2 dρ(R) of the partial isometry F onto L2 Δ(a, b) mul A and recall that

$$\ker \mathcal{F} = \text{mult} \, A = \ker \left( A - \lambda \right)^{-1}, \qquad \lambda \in \mathbb{C} \mid \mathbb{R}.$$

Then Fop is an isometry and (B.2.8) leads to

$$(A\_{\mathrm{op}} - \lambda)^{-1} = \mathcal{F}\_{\mathrm{op}}^\*(Q - \lambda)^{-1} \mathcal{F}\_{\mathrm{op}}, \qquad \lambda \in \mathbb{C} \ \backslash \mathbb{R}.$$

In general, here the restricted Fourier transform Fop is not necessarily onto.

In order to prove that Fop is surjective one needs an additional condition:

$$\begin{array}{c} \text{for each } x\_0 \in \mathbb{R} \text{ there exists a compactly} \\ \text{supported function } f \in L^2\_\Delta(a, b) \text{ such that } (\mathcal{F}\_\text{op}f)(x\_0) \neq 0. \end{array} \tag{B.2.9}$$

For the following result, recall that it is assumed that mul A is finite-dimensional.

**Theorem B.2.3.** Let ω be a continuous 1 × 2 matrix function (t, x) → ω(t, x) on (a, b) × R whose entries are real, and assume that the conditions (B.2.2) and (B.2.9) are satisfied. Then the Fourier transform

$$f \mapsto \widehat{f}, \quad \widehat{f}(x) = \int\_{a}^{b} \omega(t, x) \Delta(t) f(t) \, dt, \quad x \in \mathbb{R},$$

extends by continuity from the compactly supported functions f ∈ L2 Δ(a, b) to a surjective partial isometry F from L2 Δ(a, b) to L2 dρ(R) with ker F = mul A, that is, the restriction

$$\mathcal{F}\_{\mathrm{op}} : L^2\_{\Delta}(a, b) \ominus \mathrm{mul} \, A \to L^2\_{d\rho}(\mathbb{R})$$

is a unitary mapping. Moreover, the self-adjoint operator Aop in L2 Δ(a, b) mul A is unitarily equivalent to the multiplication operator Q by the independent variable in L2 dρ(R) via the restricted Fourier transform Fop:

$$A\_{\rm op} = \mathcal{F}\_{\rm op}^\* Q \mathcal{F}\_{\rm op}. \tag{B.2.10}$$

Proof. Is is clear from Lemma B.2.1 that F : L2 Δ(a, b) → L2 dρ(R) is a partial isometry with ker F = mul A. One verifies in the same way as in the proof of Theorem B.1.4 that F is surjective, and hence Fop is unitary. The identity (B.2.10) follows from (B.2.8). -

In the situation of Theorem B.2.3 the inverse of the Fourier transform Fop is actually given by the reverse Fourier transform G in Lemma B.2.2, which is now a unitary mapping from L2 dρ(R) to L2 Δ(a, b) mul A. Thus, for all ϕ ∈ L2 dρ(R) one has

$$\lim\_{\eta \to \mathbb{R}} \int\_{a}^{b} \left| \Delta(t)^{\frac{1}{2}} \left( \mathcal{F}\_{\text{op}}^{-1} \varphi(t) - \int\_{\eta} \omega(t, x)^{\*} \varphi(x) \, d\rho(x) \right) \right|^{2} dt = 0.$$

## **Appendix C Sums of Closed Subspaces in Hilbert Spaces**

In this appendix the sum of closed (linear) subspaces in a Hilbert space is discussed and, in particular, conditions are given so that sums of closed subspaces are closed. There is also a brief review of the opening and gap of closed subspaces.

In the following M and N will be closed subspaces of a Hilbert space H. The first lemma on the sum of closed subspaces is preliminary.

**Lemma C.1.** Let M and N be closed subspaces of H. Then the following statements are equivalent:

$$\rho\sqrt{\|f\|^2 + \|g\|^2} \le \|f + g\|, \quad f \in \mathfrak{M}, \ g \in \mathfrak{N}.\tag{C.1}$$

Proof. (i) ⇒ (ii) Assume that M + N is closed and M ∩ N = {0}. The projection from the Hilbert space M+N onto M parallel to N is a closed, everywhere defined operator. Hence, by the closed graph theorem, the projection is bounded and there exists C > 0 such that

$$\|f\| \le C\|f+g\|, \quad f \in \mathfrak{M}, \ g \in \mathfrak{N}.$$

Likewise, there exists D > 0 such that

$$\|g\| \le D \|f + g\|, \quad f \in \mathfrak{M}, \ g \in \mathfrak{N}.$$

A combination of these inequalities leads to

$$\left\|f\right\|^2 + \left\|g\right\|^2 \le (C^2 + D^2) \left\|f + g\right\|^2, \quad f \in \mathfrak{M}, \ g \in \mathfrak{N}.$$

By eventually enlarging C2 + D2 the inequality (C.1) follows with 0 <ρ< 1.

© The Editor(s) (if applicable) and The Author(s) 2020 J. Behrndt et al., *Boundary Value Problems, Weyl Functions, and Differential Operators*, Monographs in Mathematics 108, https://doi.org/10.1007/978-3-030-36714-5

(ii) ⇒ (i) Let (hn) be a sequence in M + N converging to h ∈ H. Decompose each element hn as

$$h\_n = f\_n + g\_n, \quad f\_n \in \mathfrak{M}, \quad g\_n \in \mathfrak{N}.$$

Since (hn) is a Cauchy sequence in H it follows from (C.1) that (fn) and (gn) are Cauchy sequences in M and N, respectively. Therefore, there exist elements f ∈ M and g ∈ N, so that fn → f in M and gn → g in N. Hence, h = f + g ∈ M + N. Thus, M + N is closed. To see that the sum is direct, assume that h ∈ M ∩ N. Then, in particular, h ∈ M and −h ∈ N and the inequality (C.1) with f = h and g = −h implies h = 0. -

Let M and N be closed subspaces of H. Then the intersection M ∩ N is a closed subspace which generates an orthogonal decomposition of the Hilbert space:

$$
\mathfrak{H} = (\mathfrak{M} \cap \mathfrak{M})^{\perp} \oplus (\mathfrak{M} \cap \mathfrak{N})^{\perp}
$$

In order to study properties of the sums M + N and M⊥ + N⊥ it is sometimes useful to reduce to a direct sum.

**Lemma C.2.** Let M and N be closed subspaces of H. Then the subspaces

$$
\mathfrak{M} \cap (\mathfrak{M} \cap \mathfrak{N})^\perp \quad \text{and} \quad \mathfrak{M} \cap (\mathfrak{M} \cap \mathfrak{N})^\perp
$$

have a trivial intersection and due to

$$[(\mathfrak{M}\cap\mathfrak{N})^{\perp}=[\mathfrak{M}\cap(\mathfrak{M}\cap\mathfrak{N})^{\perp}]\oplus\mathfrak{M}^{\perp}=[\mathfrak{N}\cap(\mathfrak{M}\cap\mathfrak{N})^{\perp}]\oplus\mathfrak{N}^{\perp}\tag{C.2}$$

their orthogonal complements in the subspace (M ∩ N)⊥ coincide with M⊥ and N⊥, respectively. Moreover,

$$\begin{aligned} \mathfrak{M} &= [\mathfrak{M} \cap (\mathfrak{M} \cap \mathfrak{M})^\perp] \oplus (\mathfrak{M} \cap \mathfrak{M}), \\ \mathfrak{M} &= [\mathfrak{M} \cap (\mathfrak{M} \cap \mathfrak{M})^\perp] \oplus (\mathfrak{M} \cap \mathfrak{M}), \end{aligned} \tag{C.3}$$

and, consequently, M + N has the decomposition

$$\mathfrak{M} + \mathfrak{N} = \left[ \mathfrak{M} \cap (\mathfrak{M} \cap \mathfrak{N})^\perp + \mathfrak{N} \cap (\mathfrak{M} \cap \mathfrak{N})^\perp \right] \oplus (\mathfrak{M} \cap \mathfrak{N}).\tag{C.4}$$

Proof. It is clear that M∩(M∩N)⊥ and N∩(M∩N)⊥ have a trivial intersection. To see the first identity in (C.2) note that

$$(\mathfrak{M}\cap(\mathfrak{M}\cap\mathfrak{N})^\perp \subset (\mathfrak{M}\cap\mathfrak{N})^\perp \quad \text{and} \quad \mathfrak{M}^\perp \subset (\mathfrak{M}\cap\mathfrak{N})^\perp,$$

so that the right-hand side is contained in the left-hand side. To see the other inclusion, decompose H as

$$
\mathfrak{H} = \mathfrak{M} \oplus \mathfrak{M}^{-}.
$$

Let f ∈ (M ∩ N)⊥. Then according to this decomposition one has

$$f = g + h, \quad g \in \mathfrak{M}, \quad h \in \mathfrak{M}^{\perp}.$$

Since M⊥ ⊂ (M ∩ N)⊥ one sees that g ∈ M ∩ (M ∩ N)⊥. Hence, the left-hand side is contained in the right-hand side. The second identity in (C.2) follows by interchanging M and N.

Since M∩N is a closed subspace, it is clear that H has the following orthogonal decomposition

$$
\mathfrak{H} = (\mathfrak{M} \cap \mathfrak{M})^{\perp} \oplus (\mathfrak{M} \cap \mathfrak{N})^{\perp}
$$

Hence, in an analogous way as above, the decompositions (C.3) are clear. Thus, (C.4) follows. -

The above reduction process will play a role in the following theorem.

**Theorem C.3.** Let M and N be closed subspaces of H. Then the following statements are equivalent:

Furthermore, the following statements are equivalent:

(iii) M + N is closed and M ∩ N = {0};

(iv) M⊥ + N⊥ = H,

and the following statements are equivalent:

Proof. (iii) ⇒ (iv) Assume that M∩N = {0}. It will be shown that H = M⊥ +N⊥ or, equivalently, that H ⊂ M⊥ +N⊥. To see this, choose h ∈ H. The element h ∈ H induces two linear functionals F and G on M + N by

$$F(\gamma) = (\alpha, h), \quad G(\gamma) = (\beta, h), \quad \gamma = \alpha + \beta, \quad \alpha \in \mathfrak{M}, \ \beta \in \mathfrak{M}.\tag{C.5}$$

Since M + N is closed and M ∩ N = {0}, it follows from the inequality (C.1) that the functionals F and G are bounded on M + N. Extend the functionals F and G trivially to bounded linear functionals on all of H, and denote the extensions by F and G, respectively. By the Riesz representation theorem there exist unique elements f,g ∈ H such that

$$F(\gamma) = (\gamma, f), \quad G(\gamma) = (\gamma, g), \quad \gamma \in \mathfrak{H}.\tag{C.6}$$

The definition in (C.5) implies that F(γ)=0, γ ∈ N, and G(γ) = 0, γ ∈ M. For the corresponding elements f and g in (C.6) this means that f ∈ N⊥ and g ∈ M⊥. Now for γ = α + β, α ∈ M, β ∈ N, it follows from (C.5) and (C.6) that

$$f(\gamma, h) = (\alpha, h) + (\beta, h) = F(\gamma) + G(\gamma) = (\gamma, f) + (\gamma, g).$$

This implies that k = h − f − g ∈ (M + N)⊥ ⊂ M⊥ and hence h = f + g + k with f ∈ N⊥ and g + k ∈ M⊥. Therefore, H ⊂ M⊥ + N⊥, which completes the proof.

(i) ⇒ (ii) Use the reduction process from Lemma C.2. Since M+ N is assumed to be closed it follows from (C.4) that

$$
\Delta \mathfrak{M} \cap (\mathfrak{M} \cap \mathfrak{N})^\perp + \mathfrak{N} \cap (\mathfrak{M} \cap \mathfrak{N})^\perp
$$

is closed in (M ∩ N)⊥. Since this is a direct sum, the sum of their orthogonal complements is closed in (M∩N)⊥ by the implication (iii) ⇒ (iv). Recall that the sum of their orthogonal complements coincides with M⊥ + N⊥.

(ii) ⇒ (i) This follows from the previous implication by symmetry.

(iv) ⇒ (iii) Apply (ii) ⇒ (i) to conclude that M + N is closed. This sum is direct since {0} = (M⊥ + N⊥)⊥ by assumption.

(v) ⇔ (vi) It suffices to show (v) ⇒ (vi). Note that it follows from (iii) ⇒ (iv) that M⊥ + N⊥ = H. Moreover, M⊥ ∩ N⊥ = (M + N)⊥ = {0} by assumption, which completes the argument. -

Let again M and N be closed subspaces of H and assume that M + N is closed. It follows from (C.3) that M + N can be written as

$$\begin{split} \mathfrak{M} + \mathfrak{N} &= \left( [\mathfrak{M} \cap (\mathfrak{M} \cap \mathfrak{N})^\perp] \oplus (\mathfrak{M} \cap \mathfrak{N}) \right) + \mathfrak{N} \\ &= [\mathfrak{M} \cap (\mathfrak{M} \cap \mathfrak{N})^\perp] + \mathfrak{N} .\end{split} \tag{C.7}$$

Since M + N is closed, the orthogonal decomposition H = (M + N)⊥ ⊕ (M + N) and (C.7) show that

$$\begin{aligned} \mathfrak{H} &= \left( (\mathfrak{M} + \mathfrak{M})^\perp \oplus [\mathfrak{M} \cap (\mathfrak{M} \cap \mathfrak{M})^\perp] \right) + \mathfrak{M} \\ \mathfrak{H} &= (\mathfrak{M} + \mathfrak{M})^\perp \oplus \left( [\mathfrak{M} \cap (\mathfrak{M} \cap \mathfrak{M})^\perp] + \mathfrak{M} \right) \end{aligned} \tag{C.8}$$

The sum in the identity (C.8) is direct. To see this, assume that

$$f + g + \varphi = 0, \quad f \in (\mathfrak{M} + \mathfrak{N})^\perp, \quad g \in \mathfrak{M} \cap (\mathfrak{M} \cap \mathfrak{N})^\perp, \quad \varphi \in \mathfrak{N}.$$

Then f = −g − ϕ, where −g − ϕ ∈ M+ N. Hence, f = 0 and g = −ϕ implies that g = 0 and ϕ = 0.

The decompositions (C.7) and (C.8) will be used in the proof of the next lemma.

**Lemma C.4.** Let B ∈ **B**(H, K) have a closed range and let M be a closed subspace of H. Then M + ker B is closed if and only if B(M) is closed.

Proof. Assume that M + ker B is closed and set N = ker B. It follows from the direct sum decomposition (C.8) that B maps

$$(\mathfrak{M} + \mathfrak{M})^\perp \oplus [\mathfrak{M} \cap (\mathfrak{M} \cap \mathfrak{M})^\perp]$$

bijectively onto the closed subspace ran B and hence B also provides a bijection between the closed subspaces of (M + N)⊥ ⊕ [M ∩ (M ∩ N)⊥] and those of ran B. In particular, it follows from this observation and (C.7) that

$$B(\mathfrak{M}) = B\left(\mathfrak{M} \cap (\mathfrak{M} \cap \mathfrak{M})^\perp\right),$$

is closed in K.

For the converse statement assume that B(M) is closed. In order to show that M + ker B is closed, let fn ∈ M and ϕn ∈ ker B have the property that

$$f\_n + \varphi\_n \to \chi$$

for some χ ∈ H. In particular, this shows that Bfn → Bχ. The assumption that B(M) is closed implies that Bfn → Bf for some f ∈ M. Hence,

$$
\chi - f = \varphi \in \ker B \quad \text{or} \quad \chi = f + \varphi \in \mathfrak{M} + \ker B.
$$

It follows that M + ker B is closed. -

There is another way to approach the topic of sums of closed subspaces, namely via various notions of opening or gap between closed subspaces.

**Definition C.5.** Let M and N be closed subspaces of H with corresponding orthogonal projections PM and PN, respectively. The opening ω(M, N) between M and N is defined as

$$\omega(\mathfrak{M}, \mathfrak{N}) = \|P\_{\mathfrak{M}} P\_{\mathfrak{N}}\|\,.$$

It is clear that 0 ≤ ω(M, N) ≤ 1 and that ω(M, N) = ω(N,M), since one has A = A∗ for any A ∈ **B**(H).

**Proposition C.6.** Let M and N be closed subspaces of H. Then the following statements are equivalent:

Proof. (i) ⇒ (ii) Assume that ω(M, N) < 1. Observe that for f ∈ M and g ∈ N one has

$$|(f,g)| = |(P\_{\mathfrak{M}}f, P\_{\mathfrak{N}}g)| \le \omega(\mathfrak{M}, \mathfrak{N}) ||f|| ||g||.$$

It follows that for all f ∈ M and g ∈ N

$$\begin{aligned} \|f\|^2 + \|g\|^2 &= \|f + g\|^2 - 2\text{Re}\,(f, g) \\ &\le \|f + g\|^2 + 2|\langle f, g\rangle| \\ &\le \|f + g\|^2 + 2\omega(\mathfrak{M}, \mathfrak{N}) \|f\| \|g\| \\ &\le \|f + g\|^2 + \omega(\mathfrak{M}, \mathfrak{N}) \left(\|f\|^2 + \|g\|^2\right). \end{aligned}$$

In particular, this shows that

$$(1 - \omega(\mathfrak{M}, \mathfrak{N})) \left( \|f\|^2 + \|g\|^2 \right) \le \|f + g\|^2, \quad f \in \mathfrak{M}, \quad g \in \mathfrak{N}.$$

Hence, Lemma C.1 implies that M ∩ N = {0} and that M + N is closed, which gives (ii).

(ii) ⇒ (i) Assume that M + N is closed and M ∩ N = {0}. By Lemma C.1, the inequality (C.1) holds for some 0 <ρ< 1, hence for all f ∈ M and g ∈ N

$$\rho^2 \left( \|f\|^2 + \|g\|^2 \right) \le \|f\|^2 + 2\text{Re}\left(f, g\right) + \|g\|^2$$

or, equivalently,

$$-2\text{Re}\left(f,g\right)\le\left(1-\rho^2\right)\left(\|f\|^2+\|g\|^2\right).\tag{C.9}$$

Now let h, k ∈ H with h ≤ 1 and k ≤ 1 and choose θ ∈ R such that

$$|e^{i\theta}(P\_{\mathfrak{M}}h, P\_{\mathfrak{N}}k) = -|(P\_{\mathfrak{M}}h, P\_{\mathfrak{N}}k)|.$$

Then (eiθPMh, PNk) ∈ R and (C.9) with f = eiθPMh and g = PNk yields

$$\begin{aligned} |(h, P\_{\mathfrak{M}} P\_{\mathfrak{N}} k)| &= |(P\_{\mathfrak{M}} h, P\_{\mathfrak{N}} k)| \\ &= -\operatorname{Re}\left(e^{i\theta} P\_{\mathfrak{M}} h, P\_{\mathfrak{N}} k\right) \\ &\leq \frac{1 - \rho^2}{2} \left( ||e^{i\theta} P\_{\mathfrak{M}} h||^2 + ||P\_{\mathfrak{N}} k||^2 \right) \\ &\leq 1 - \rho^2. \end{aligned}$$

This implies PMPN ≤ 1 − ρ2 < 1 and hence (i) holds. -

**Corollary C.7.** Let M and N be closed subspaces of H such that M + N is closed and M ∩ N = {0}. Then for closed subspaces M1 ⊂ M and N1 ⊂ N also the subspace M1 + N1 is closed.

Proof. This statement follows immediately from Proposition C.6 by noting that ω(M1, N1) ≤ ω(M, N) < 1. -

Proposition C.6 and the reduction of closed subspaces in Lemma C.2 leads to the following corollary.

**Corollary C.8.** Let M and N be closed subspaces of H. Then the following statements are equivalent:

Proof. Since the closed subspaces M ∩ (M ∩ N)⊥ and N ∩ (M ∩ N)⊥ have trivial intersection, it follows from Proposition C.6 that (i) is equivalent to

$$\mathfrak{M} \cap (\mathfrak{M} \cap \mathfrak{N})^\perp + \mathfrak{M} \cap (\mathfrak{M} \cap \mathfrak{N})^\perp \tag{C.10}$$

is closed. Furthermore, the decomposition (C.4) shows that the space in (C.10) is closed if and only if M + N is closed. -

The notion of opening is now supplemented with the notion of gap between closed subspaces.

**Definition C.9.** Let M and N be closed subspaces of H with corresponding orthogonal projections PM and PN, respectively. The gap g(M, N) between M and N is defined as

$$g(\mathfrak{M}, \mathfrak{N}) = ||P\_{\mathfrak{M}} - P\_{\mathfrak{N}}||.$$

Note that it follows directly from the definition that

$$g(\mathfrak{M}, \mathfrak{N}) = g(\mathfrak{N}, \mathfrak{M}) \quad \text{and} \quad g(\mathfrak{M}^\perp, \mathfrak{N}^\perp) = g(\mathfrak{M}, \mathfrak{N}).\tag{C.11}$$

The connection between the gap and the opening between closed subspaces is contained in the following proposition.

**Proposition C.10.** Let M and N be closed subspaces in H. Then

$$g(\mathfrak{M}, \mathfrak{N}) = \max\left\{\omega(\mathfrak{M}, \mathfrak{N}^{\perp}), \omega(\mathfrak{M}^{\perp}, \mathfrak{N})\right\} \tag{C.12}$$

and, in particular, g(M, N) ≤ 1.

Proof. First observe that

$$P\_{\mathfrak{N}^\perp}P\_{\mathfrak{N}} = (I - P\_{\mathfrak{N}})P\_{\mathfrak{N}} = (P\_{\mathfrak{N}} - P\_{\mathfrak{N}})P\_{\mathfrak{N}^\perp}$$

so that ω(M, N⊥) = ω(N⊥,M) ≤ g(M, N) PM ≤ g(M, N). Therefore, also

$$\omega(\mathfrak{M}^\perp, \mathfrak{N}) \le g(\mathfrak{M}^\perp, \mathfrak{N}^\perp) = g(\mathfrak{M}, \mathfrak{N})$$

and hence

$$\max\left\{\omega(\mathfrak{M},\mathfrak{N}^\perp), \omega(\mathfrak{M}^\perp,\mathfrak{N})\right\} \le g(\mathfrak{M},\mathfrak{N}).$$

Furthermore, observe that for all h ∈ H

$$\begin{aligned} \|(P\_{\mathfrak{M}} - P\_{\mathfrak{N}})h\|^2 &= \|(I - P\_{\mathfrak{N}})P\_{\mathfrak{M}}h - P\_{\mathfrak{N}}(I - P\_{\mathfrak{M}})h\|^2 \\ &= \|P\_{\mathfrak{N}^\perp}P\_{\mathfrak{M}}h\|^2 + \|P\_{\mathfrak{N}}P\_{\mathfrak{M}^\perp}h\|^2 \\ &= \|P\_{\mathfrak{N}^\perp}P\_{\mathfrak{M}}P\_{\mathfrak{M}}h\|^2 + \|P\_{\mathfrak{N}}P\_{\mathfrak{M}^\perp}P\_{\mathfrak{M}^\perp}h\|^2 \\ &\leq \omega(\mathfrak{M}, \mathfrak{N}^\perp)^2 \|P\_{\mathfrak{M}}h\|^2 + \omega(\mathfrak{M}^\perp, \mathfrak{N})^2 \|P\_{\mathfrak{M}^\perp}h\|^2, \end{aligned}$$

and the last term is majorized by

$$\max\left\{\omega(\mathfrak{M},\mathfrak{N}^\perp)^2, \omega(\mathfrak{M}^\perp,\mathfrak{N})^2\right\} \left(\|P\_{\mathfrak{M}}h\|^2 + \|P\_{\mathfrak{M}^\perp}h\|^2\right).$$

It follows that

$$g(\mathfrak{M}, \mathfrak{N}) \le \max\left\{\omega(\mathfrak{M}, \mathfrak{N}^\perp), \omega(\mathfrak{M}^\perp, \mathfrak{N})\right\}.$$

Therefore, (C.12) has been established. -

**Proposition C.11.** Let M and N be closed subspaces of H. Then

g(M, N) < 1 ⇒ dimM = dim N.

Proof. If g(M, N) = PM − PN < 1, then the bounded operator

$$I - (P\mathfrak{m} - P\mathfrak{n})$$

maps H onto itself with bounded inverse. Hence,

$$P\_{\mathfrak{M}}P\_{\mathfrak{N}}\mathfrak{H} = P\_{\mathfrak{M}}\left[I - (P\_{\mathfrak{M}} - P\_{\mathfrak{N}})\right]\mathfrak{H} = P\_{\mathfrak{M}}\mathfrak{H},$$

so that PM maps ran PN onto ran PM. Observe that for h ∈ ran PN

$$\|\|P\_{\mathfrak{M}}h\|\| = \|h + (P\_{\mathfrak{M}} - P\_{\mathfrak{N}})h\|\| \ge \left(1 - \|\|P\_{\mathfrak{M}} - P\_{\mathfrak{N}}\|\|\right) \|h\|\|$$

and hence PM maps ran PN boundedly and boundedly invertible onto ran PM. This implies that the dimensions of M = ran PM and N = ran PN coincide. -

**Theorem C.12.** Let M and N be closed subspaces of H. Then the following statements are equivalent:

and in this case dimM = dim N⊥ and dimM⊥ = dim N.

Proof. (i) ⇒ (ii) Proposition C.10 implies ω(M, N) < 1 and ω(M⊥, N⊥) < 1. By Proposition C.6, the condition ω(M, N) < 1 implies that M+N is closed and that M∩N = {0}, and in the same way ω(M⊥, N⊥) < 1 yields that M⊥ +N⊥ is closed and that M⊥ ∩ N⊥ = {0}. The identity M⊥ ∩ N⊥ = {0} implies that M + N is dense in H. Therefore, M + N = H.

(ii) ⇒ (i) It follows from Proposition C.6 that ω(M, N) < 1. Moreover, the equivalence of (v) and (vi) in Theorem C.3 shows M⊥ + N⊥ = H and M⊥ ∩ N⊥ = {0}, and therefore ω(M⊥, N⊥) < 1 by Proposition C.6. Now Proposition C.10 implies that g(M, N⊥) < 1.

It is clear from (i) and Proposition C.11 that dimM = dim N⊥. Furthermore, as g(M⊥, N) = g(M, N⊥) by (C.11), the same argument gives dim M⊥ = dim N. -

## **Appendix D**

## **Factorization of Bounded Linear Operators**

This appendix contains a number of results pertaining to the factorization of bounded linear operators based on range inclusions or norm inequalities. These results will be useful in conjunction with range inclusions or norm inequalities for relations.

Let H and K be Hilbert spaces and let H ∈ **B**(H, K). The restriction of H to (ker H)⊥ ⊂ H,

$$H: \text{ (ker } H)^\perp \to \text{ran } H,$$

is a bijective mapping between (ker H)⊥ and ran H. The inverse of this restriction is a (linear) operator from ran H onto (ker H)⊥ and is called the Moore–Penrose inverse of the operator H, which in general is an unbounded operator. Since H = ker H ⊕ (ker H)⊥, one sees immediately that

$$H^{(-1)}H = P\_{(\ker H)^\perp} \,. \tag{D.1}$$

Moreover, the closed graph theorem shows that ran H is closed if and only if H(−1) takes ran H boundedly into (ker H)⊥. Hence, if ran H is closed, then

$$H^{(-1)} \in \mathbf{B}\left(\operatorname{ran} H, (\ker H)^{\perp}\right) \quad \text{and} \quad (H^{(-1)})^\times \in \mathbf{B}\left((\ker H)^{\perp}, \operatorname{ran} H\right), \tag{D.2}$$

where × denotes the adjoint with respect to the scalar products in ran H and (ker H)⊥.

**Lemma D.1.** Let H and K be Hilbert spaces and let H ∈ **B**(H, K). Then ran H is closed if and only if ran H∗ is closed.

Proof. Since H ∈ **B**(H, K), it suffices to show that if ran H is closed in K, then ran H∗ is closed in H. Since ran H∗ = (ker H)⊥, one only needs to show

$$(\ker H)^\perp \subset \operatorname{ran} H^\*.$$

J. Behrndt et al., *Boundary Value Problems, Weyl Functions, and Differential Operators*, Monographs in Mathematics 108, https://doi.org/10.1007/978-3-030-36714-5

© The Editor(s) (if applicable) and The Author(s) 2020

For this let h ∈ (ker H)⊥ and f ∈ H. Then it follows from (D.1) and (D.2) that with (H(−1))× ∈ **B**((ker H)⊥,ran H) in (D.2) one has

$$\begin{aligned} (h, f)\_{\mathfrak{H}} &= (h, P\_{\text{(ker } H)^\perp} f)\_{\text{(ker } H)^\perp} \\ &= \left( h, H^{(-1)} H f \right)\_{\text{(ker } H)^\perp} \\ &= \left( (H^{(-1)})^\times h, H f \right)\_{\text{ran } H} \\ &= \left( (H^{(-1)})^\times h, H f \right)\_{\mathfrak{H}} \\ &= \left( H^\* (H^{(-1)})^\times h, f \right)\_{\mathfrak{H}} \end{aligned}$$

and it follows that h = H∗(H(−1))×h. Hence, h ∈ ran H∗ and therefore ran H∗ is closed. -

Lemma D.1 has a direct consequence, which is useful.

**Lemma D.2.** For H ∈ **B**(H, K) there are the inclusions

$$
\overline{\text{ran}}\,HH^\* \subset \text{ran}\,H \subset \overline{\text{ran}}\,H = \overline{\text{ran}}\,HH^\*.\tag{D.3}
$$

Moreover, the following statements are equivalent:

in which case all inclusions in (D.3) are identities.

Proof. The chain of inclusions in (D.3) is clear; the last equality is a consequence of the identity ker H∗ = ker HH∗.

(i) ⇒ (iii) Assume that ran HH∗ is closed. Then all inclusions in (D.3) are equalities and, in particular, (iii) follows.

(iii) ⇒ (ii) Assume that ran HH∗ = ran H. It will be sufficient to show that ran H∗ ⊂ ran H∗, which implies that ran H∗ is closed and hence also ran H is closed by Lemma D.1. Assume that h ∈ ran H∗ = (ker H)⊥. By the assumption, it follows that Hh = HH∗k for some k, which gives H(h − H∗k) = 0. Note that h ∈ (ker H)⊥ and H∗k ∈ ran H∗ ⊂ ran H∗ = (ker H)⊥, and hence h = H∗k. Thus, ran H∗ ⊂ ran H∗ which implies that ran H∗ is closed, so that also ran H is closed by Lemma D.1.

(ii) ⇒ (i) Assume that ran H is closed. Then it follows from (D.3) that

$$
\overline{\text{ran}}\,HH^\* = \text{ran}\,H.\tag{\text{D.4}}
$$

It will suffice to show that ran HH∗ ⊂ ran HH∗. Assume that k ∈ ran HH∗. Then k = Hh for some h ∈ (ker H)⊥ by (D.4). As (ker H)⊥ = ran H∗ = ran H∗ by the assumption (ii) and Lemma D.1, one has h ∈ ran H∗ and therefore k ∈ ran HH∗. This shows that ran HH∗ ⊂ ran HH∗ which implies that ran HH∗ is closed. -

Let F, G, and H be Hilbert spaces, let A ∈ **B**(F, H), B ∈ **B**(G, H), and C ∈ **B**(F, G), and assume that the following factorization holds:

$$A = BC.\tag{D.5}$$

Then it follows from (D.5) that ker C ⊂ ker A. Note that the orthogonal decomposition G = ker B⊕ran B∗ allows one to write A = BPC, where P is the orthogonal projection in G onto ran B∗. Hence, one may always assume that ran C ⊂ ran B∗. In this case ker C = ker A and hence C maps (ker A)⊥ = ran A∗ into ran B∗ injectively. Moreover, in this case the operator C in (D.5) is uniquely determined. The following proposition is a version of the well-known Douglas lemma.

**Proposition D.3.** Assume that A ∈ **B**(F, H) and B ∈ **B**(G, H). Let ρ > 0, then the following statements are equivalent:

Moreover, if ran A ⊂ ran B, then B(−1)ϕ ≤ ρ A(−1)ϕ , ϕ ∈ ran A, for some ρ > 0.

Proof. (i) ⇒ (ii) It is clear that ran A ⊂ ran B and therefore also

$$B^{(-1)}A = B^{(-1)}B \\ C = P\_{(\ker B)^\perp} \\ C = P\_{\overline{\mathtt{T}} \mathtt{T} \mathtt{B}^\*} \\ C = C.$$

From the last identity one sees that

$$\|\|B^{(-1)}A\psi\|\| \le \|C\| \|\|\psi\|\|, \quad \psi \in \mathfrak{F}.$$

Now let ϕ ∈ ran A. Then there exists ψ ⊥ ker A such that ϕ = Aψ. Therefore, ψ = A(−1)ϕ and it follows immediately that

$$\|\|B^{(-1)}\varphi\|\| \le \|C\| \|\|A^{(-1)}\varphi\|\|, \quad \varphi \in \text{ran}\,A.$$

Hence, (ii) has been shown.

(ii) ⇒ (i) It follows from (ii) that for each h ∈ F there exists a uniquely determined element k ∈ (ker B)⊥ = ran B∗ such that Ah = Bk. Hence, the mapping h → k from F to ran B∗ ⊂ G defines a linear operator C with dom C = F and one has A = BC. To show that the operator C is closed, assume that

$$h\_n \to h \quad \text{and} \quad k\_n = Ch\_n \to k,$$

where kn ∈ ran B∗. Since A and B are bounded linear operators, it follows from from Ahn = Bkn that Ah = Bk. Furthermore, ran B∗ is closed, which implies that k ∈ ran B∗. Hence, k = Ch and thus C is closed. By the closed graph theorem, it follows that C ∈ **B**(F, G). The property ran C ⊂ ran B∗ holds by construction. Furthermore, one sees that for ψ ∈ F

$$\begin{aligned} \|C\psi\| &= \|P\_{\text{ran}\,B^\*}C\psi\| \\ &= \|P\_{(\text{ker}\,B)^\perp}C\psi\| \\ &= \|B^{(-1)}BC\psi\| \\ &\le \rho\,\|A^{(-1)}A\psi\| \\ &= \rho\,\|P\_{(\text{ker}\,A)^\perp}\psi\| \\ &\le \rho\,\|\psi\|, \end{aligned}$$

and so C ≤ ρ.

(i) ⇒ (iii) It is clear that AA∗ = BCC∗B∗. For ψ ∈ F one has

$$\left(AA^\*\psi,\psi\right) = \left\|C^\*B^\*\psi\right\|^2 \le \left\|C^\*\right\|^2 \left\|B^\*\psi\right\|^2 = \left\|C^\*\right\|^2 \left(BB^\*\psi,\psi\right)$$

and hence AA∗ ≤ C∗ 2BB∗. Since C∗ = C ≤ ρ this leads to (iii).

(iii) ⇒ (i) The mapping D from ran B∗ to ran A∗ given by

$$B^\*h \mapsto A^\*h, \quad h \in \mathfrak{H},$$

is well defined and linear. To see that it is well defined observe that B∗h = 0 implies AA∗h = 0 and hence A∗h = 0. Then DB∗h = A∗h for h ∈ H and one has D ∈ **B**(ran B∗,ran A∗) with DB∗h ≤ ρ2 B∗h . Due to the boundedness, this operator has a unique extension in **B**(ran B∗,ran A∗), denoted again by D, and D ≤ ρ2. Finally, this mapping has a trivial extension D ∈ **B**(G, F), satisfying DB∗h = A∗h for h ∈ H and again D ≤ ρ2. With C = D∗ one obtains that C ∈ **B**(F, G), A = BC, and C ≤ ρ2. In view of the construction of D one has ker B = (ran B∗)⊥ ⊂ ker D, so that ran C = ran D∗ ⊂ ran B∗.

For the last statement it suffices to observe that the inclusion ran A ⊂ ran B implies A = BC for some C ∈ **B**(F, G) (see (ii) ⇒ (i)). However, this implies AA∗ ≤ ρ2 BB∗ for some ρ > 0 (see (i) ⇒ (iii)). -

The next result contains a strengthening of Proposition D.3. Recall that ran C ⊂ ran B∗ implies that ker C = ker A. Hence, if C maps ran A∗ onto ran B∗, then C maps ran A∗ onto ran C = ran B∗ bijectively.

**Corollary D.4.** Assume that A ∈ **B**(F, H) and B ∈ **B**(G, H). Let ρ > 0. Then the following statements are equivalent:

(a) A = BC for some C ∈ **B**(F, G) with ran C = ran B∗ and C ≤ ρ;

(b) ran A = ran B and B(−1)ϕ ≤ ρ A(−1)ϕ , ϕ ∈ ran A;

(c) ε2BB∗ ≤ AA∗ ≤ ρ2 BB∗ for some 0 <ε<ρ.

Moreover, if ran A = ran B, then B(−1)ϕ ≤ ρ A(−1)ϕ , ϕ ∈ ran A, for some ρ > 0.

Proof. The assertions in Proposition D.3 will be freely used in the arguments below.

(a) ⇒ (c) Recall that C maps ran A∗ bijectively onto ran B∗. Since

ran C∗ = ran A∗ and ker C∗ = ker B,

one sees that C∗ maps ran B∗ bijectively onto ran A∗, as ran C∗ is closed by Lemma D.1. Thus, CC∗ is a bijective mapping from ran B∗ onto itself and hence

$$(AA^\*h, h) = (CC^\*B^\*h, B^\*h) \ge \varepsilon^2 (B^\*h, B^\*h) = \varepsilon^2 (BB^\*h, h), \quad h \in \mathfrak{H}.$$

Therefore, (c) follows.

(c) ⇒ (b) The inequality BB∗ ≤ ε−2AA∗ implies ran B ⊂ ran A.

(b) ⇒ (a) It suffices to show that ran B∗ ⊂ ran C. Let k ∈ ran B∗. The assumption ran B ⊂ ran A implies that Bk = Ah for some h ∈ F. Since A = BC, one sees that Bk = BCh, and hence k − Ch ∈ ker B. On the other hand, k ∈ ran B∗ and Ch ∈ ran C ⊂ ran B∗ yield k − Ch ∈ ran B∗ = (ker B)⊥. Therefore, k = Ch and it follows that ran B∗ ⊂ ran C holds. -

An operator T ∈ **B**(F, G) is said to be a partial isometry if the restriction of T to (ker T)⊥ is an isometry. The initial space is (ker T)⊥ and the final space is ran T, which is automatically closed since (ker T)⊥ is closed and the restriction of T is isometric. Let T ∈ **B**(F, G), then the following statements are equivalent:

The next result can be considered to be a special case of Proposition D.3 and Corollary D.4.

**Corollary D.5.** Assume that A ∈ **B**(F, H) and B ∈ **B**(G, H). Then the following statements are equivalent:

Proof. (i) ⇒ (ii) Observe that AA∗ = BCC∗B∗. By assumption, C is a partial isometry with initial space ran A∗ and final space ran B∗. Therefore, C∗ is a partial isometry with initial space ran B∗ and final space ran A∗, and it follows that CC∗ is the orthogonal projection onto the final space ran B∗ of C. Hence, AA∗ = BB∗.

(ii) ⇒ (i) It follows from Corollary D.4 that A = BC for some C ∈ **B**(F, G) with ran C = ran B∗ and C ≤ 1. Moreover

$$\| |C^\*B^\*h| \| = \| A^\*h \| = \| B^\*h \|, \qquad h \in \mathfrak{H},$$

shows that C∗ is a partial isometry with initial space ran B∗ and final space ran A∗. Therefore, C is a partial isometry with initial space ran A∗ and final space ran B∗. -

The following polar decomposition of H ∈ **B**(F, H) can be seen as a consequence of the previous corollaries. Recall that |H∗| ∈ **B**(H) is a nonnegative operator in H defined by |H∗| = (HH∗) 1 2 .

**Corollary D.6.** Let H ∈ **B**(F, H). Then H = |H∗|C and |H∗| = HC∗ for some partial isometry C ∈ **B**(F, H) with initial space ran H∗ and final space ran |H∗|. In addition, one has ran H = ran |H∗|.

The following corollary is a direct consequence of Lemma D.2.

**Corollary D.7.** Let H ∈ **B**(H) be self-adjoint and nonnegative and let H 1 2 be its square root. Then there are the inclusions

$$
\operatorname{ran} H \subset \operatorname{ran} H^{\frac{1}{2}} \subset \overline{\operatorname{ran}} H^{\frac{1}{2}} = \overline{\operatorname{ran}} H. \tag{D.6}
$$

Moreover, the following statements are equivalent:

in which case all the inclusions in (D.6) are identities.

Recall that the Moore–Penrose inverse of H 1 2 is the uniquely defined inverse of the restriction

$$H^{\frac{1}{2}} \colon (\ker H^{\frac{1}{2}})^\perp = (\ker H)^\perp \to \text{ran} \, H^{\frac{1}{2}} \dots$$

In the sequel the Moore–Penrose inverse of H 1 2 will be denoted by H(− 1 2 ) , i.e.,

$$H^{(-\frac{1}{2})} := (H^{\frac{1}{2}})^{(-1)} .$$

It is clear that

$$H^{( -\frac{1}{2})} \in \mathbf{B}(\text{ran}\,H^{\frac{1}{2}}, (\text{ker}\,H)^{\perp}) \quad \Leftrightarrow \quad \text{ran}\,H^{\frac{1}{2}} \text{ is closed}.$$

Here is the version of Proposition D.3 for nonnegative operators.

**Proposition D.8.** Let A, B ∈ **B**(H) be self-adjoint and nonnegative, and let ρ > 0. Then the following statements are equivalent:

Morover, if ran A1 2 ⊂ ran B 1 2 , then B(− 1 2 ) ϕ ≤ ρ A(− 1 2 ) ϕ , ϕ ∈ ran A1 2 , for some ρ > 0.

## **Notes**

These pages contain a number of comments about the various results in this monograph and some further developments. We also give some historical remarks without any claim of completeness; the history of this area is complicated, also because of the limited exchange of ideas that has existed between East and West. The main setting of the book is that of operators and relations in Hilbert spaces. For an introduction and comprehensive treatment of linear operators in Hilbert spaces and related topics we refer the reader to the standard textbooks [2, 133, 141, 272, 276, 341, 348, 598, 649, 650, 651, 652, 673, 691, 723, 738, 743, 752, 756, 757]. We shall occasionally address topics that fall outside the scope of the monograph itself, e.g., the role of operators and relations in spaces with an indefinite inner product is indicated with proper references.

#### **Notes on Chapter 1**

Linear relations in Hilbert spaces go back at least to Arens [42]; they attracted attention because of their usefulness in the extension theory of not necessarily densely defined operators; see for instance [82, 128, 202, 218, 266, 297, 523, 619, 638]. Linear relations also appear in a natural way in the description of boundary conditions; cf. [2, 660]. Related material can be found in [74, 298, 406, 512, 513, 514, 515, 516, 576, 683, 685, 686]. The description of self-adjoint, maximal dissipative, or accumulative extensions of a symmetric operator (see Sections 1.4, 1.5, and 1.6) in terms of operators between the defect spaces in Theorem 1.7.12 and Theorem 1.7.14 goes back to von Neumann and Straus [ ˇ 610, 731] in the densely defined case and was worked out in the general case by Coddington [202, 203]; see also [266, 490, 733]. These descriptions may be seen as building stones for the extension theory via boundary triplets in Chapter 2.

Sections 1.1, 1.2, and 1.3 present elementary facts. The identity (1.1.10) goes back to Straus [ ˇ 726]. The notions of spectrum and points of regular type for a linear relation in Definition 1.2.1 and Definition 1.2.3 are introduced as in the operator case [268, 269, 404], while the adjoint of a linear relation in Definition 1.3.1 is introduced as in [611], see also [42, 659] and [396, 599]. Note that in Proposition 1.2.9 one sees a "pseudo-resolvent" which already shows that there is a relation in the background; cf. [421]. In Theorem 1.3.14 we present the notion of the operator part of a relation and the corresponding Hilbert space decomposition in the spirit of the Lebesgue decomposition of a relation [397, 398]; see also [439, 621, 622, 623] for the operator case. Operator parts have a connection with generalized inverses (see for instance [127, 609]), such as the Moore–Penrose inverse in Definition 1.3.17. Note that the more general definition Pker HH−1 gives a Moore–Penrose inverse that is a closable operator. The special classes of relations and their transforms are developed in the usual way; see [660, 705, 706]. The presentation is influenced by personal communication with McKelvey (see also [572]) and lecture notes by Kaltenb¨ack [451] and Woracek [775, 776]; see also [365, 556, 662] for more references. Although we assume a working knowledge of the spectral theory of self-adjoint operators in Hilbert spaces, we highlight in Section 1.5 a couple of useful facts, with special attention paid to the semibounded case. Lemma 1.5.7 goes back to [210], Proposition 1.5.11 is a consequence of the Douglas lemma in Appendix D, and Lemma 1.5.12 is taken from [95]. The extension theory in the sense of von Neumann can be found in Section 1.7. Note that extensions of a symmetric relation may be disjoint or transversal. Our present characterization of these notions in Theorem 1.7.3 seems to be new; see also [391].

For the development of boundary triplets we only need a few basic facts concerning spaces with an indefinite inner product, see Section 1.8. A typical result in this respect is the automatic boundedness property in Lemma 1.8.1; see [73, 77, 235, 236]. The transform in Definition 1.8.4 goes back to Shmuljan [506, 705, 706]. The notions of strong graph convergence and strong resolvent convergence of relations are treated in Section 1.9 (see [650] for the case of self-adjoint operators). Here we prove the equivalence of the two notions when there is a uniform bound. For Corollary 1.9.5 see [755]. The parametric representation of linear relations in Section 1.10 is closely connected with the theory of operator ranges. Here we only consider such representations from the point of view of boundary value problems; see also [2, 660] and Chapter 2. However, these parametric representations also play an important role in the above mentioned Lebesgue decompositions of relations and, more generally, in the Lebesgue type decompositions of relations; cf. [397]. Most of the material in the beginning of Section 1.11 consists of straightforward generalizations of properties of the resolvent. Lemma 1.11.4 is well known for the usual resolvent and seems to be new in the generalized context; it is used in Section 2.8. Lemma 1.11.5 was inspired by [523]. Nevanlinna families and pairs are generalizations of operator-valued Nevanlinna functions; cf. Appendix A. The notion of Nevanlinna pair goes back to [617]. The notion of Nevanlinna family appears already in [523] (and in a different guise in [266]) and was later more systematically studied in [20, 21, 22, 89, 94, 95, 234, 246].

Although in this monograph we will restrict ourselves mainly to Hilbert spaces, we will make some comments in the notes concerning the setting of indefinite inner product spaces; see for instance [31, 77, 79, 142, 430, 452] and also [775, 776] for Pontryagin spaces, almost Pontryagin spaces, Kre˘ın spaces, and almost Kre˘ın spaces. The spectral theory of operators and relations in such spaces

has been studied extensively; in Kre˘ın spaces the operators are often required to have (locally) finitely many negative squares or to be (locally) definitizable, that is, roughly speaking some polynomial (or rational function) of the operator or relation under consideration is (locally) nonnegative in the Kre˘ın space sense. Here we only mention work related to the present topics [75, 78, 79, 99, 220, 222, 253, 260, 264, 268, 269, 274, 275, 434, 435, 436, 437, 497, 498, 499, 500, 501, 502, 503, 519, 522, 631, 632, 633, 634, 674, 717, 718, 719, 720, 721, 722, 764, 765].

#### **Notes on Chapter 2**

Boundary triplets were originally introduced in the works of Bruk [176, 177, 178] and Kochube˘ı [466] and were used in the study of operator differential equations in the monograph [346]; see also [509]. The notion of boundary triplet appeared implicitly in a different form already much earlier in the work of Calkin [186, 187] (see also [276, Chapter XII] and [410]). The Weyl function was introduced by Derkach and Malamud in [243, 244]; it can also be interpreted as the Q-function from the works of Kre˘ın [491, 492] (see also [679]) and, e.g., the papers [499, 500, 501, 502, 503, 504] of Kre˘ın and Langer, and [523] of Langer and Textorius. We also mention the more recent monographs [359, 691], where boundary triplets and Weyl functions are briefly discussed, as well as the very recent monograph by Derkach and Malamud [247] (in Russian), in which a more detailed exposition and more than 400 references (also many older papers published in Russian) can be found. From our point of view the most influential general papers on boundary triplets and Weyl functions area are [245, 246] and [184], the latter also has an introduction to the basic notions.

While a large part of the material in this chapter can be found in the literature, the presentation here is given in a unified form for general symmetric relations. The connection between the abstract Green identity and the appropriate indefinite inner products (see Section 1.8) is used in Section 2.1 when deriving some of the basic properties of boundary triplets and their transforms. The inverse result for boundary triplets in Theorem 2.1.9 is taken from [103, 104]. The discussion in Section 2.2 on parametric representations of boundary conditions is given to establish a connection between abstract and concrete boundary value problems. The definitions and the central properties of γ-fields and Weyl functions are presented in Section 2.3 and these results can be found in [245, 246]. In particular, Proposition 2.3.2 (part (ii)) and Proposition 2.3.6 (parts (iii), (v)) show the connection with the notion of Q-function (see [184, 245, 246]), while the formula in part (vi) of Proposition 2.3.6 is taken from [265]. Without specification of the γ-field and Weyl function, constructions of boundary triplets appear already in the early literature; see [176, 466, 577] and the monographs [346, 509]. Here the constructions are based on decompositions that are closely related to the von Neumann formulas in Section 1.7. The Weyl function in Theorem 2.4.1 coincides with the abstract Donoghue type M-function that was studied, for instance, in [317, 320, 327]. Transformations of boundary triplets and the corresponding

γ-fields and Weyl functions have been treated in [234, 237, 245, 246]. In particular, the description of boundary triplets in Theorem 2.5.1 is taken from [246]. We make the precise connection with Q-functions in Corollary 2.5.8. The present notion of unitary equivalence of boundary triplets in Definition 2.5.14 is slightly more general than the definition used in [382].

Section 2.6 is devoted to Kre˘ın's formula for intermediate extensions. The Kre˘ın type formula (for intermediate extensions) phrased in terms of boundary triplets in Theorem 2.6.1 is adapted to the present setting from [233]. This yields new simple proofs for the description of various parts of the spectra of intermediate extensions in Theorem 2.6.2. This description appears in a similar form in [243, 244, 245, 246] and is treated for dissipative extensions in [346, 468] by means of characteristic functions. The same remark applies to the proof of Theorem 2.6.5, which can be found in a different form in [184]. These results indicate the importance of Kre˘ın's formula for the spectral analysis of self-adjoint extensions of symmetric operators. Closely related is the completion problem, briefly touched upon in Remark 2.4.4, which is investigated in [383], where also an analogous boundary triplet appears for the nonnegative case. This case of extensions of a bounded symmetric operator was studied earlier in, for instance, [735, 736].

Section 2.7 is devoted to the Kre˘ın–Na˘ımark formula, which is our terminology for the Kre˘ın formula for compressed resolvents of self-adjoint exit space extensions, in short, Kre˘ın's formula for exit space extensions. Actually, it was Na˘ımark [605, 606, 607] who investigated different types of extensions (with exit) and a proper interpretation then yields the corresponding resolvent formula. For the notion of Straus family and for Kre˘ın's formula for exit space extensions we refer to ˇ [491, 492, 679] and to [726, 727, 728, 729, 730, 731, 732, 733]; some other related papers are [51, 121, 126, 219, 234, 237, 254, 266, 320, 497, 523, 554, 595, 596, 624]. The Straus families are restrictions of the adjoint in terms of " ˇ λ-dependent boundary conditions" given by a Nevanlinna family or corresponding Nevanlinna pair. When the exit space is a Pontryagin space, the same mechanism is in force and the boundary conditions are now given by a generalized Nevanlinna family or corresponding Nevanlinna pair (generalized in the sense that the family or pair has negative squares); we mention [76, 100, 111, 253, 257, 258, 259, 260, 261, 263, 290, 521, 523] and [664] for the special case of a finite-dimensional exit space. Boundary triplets for symmetric relations in Pontryagin and even in Kre˘ın spaces were introduced by Derkach (see [101, 227, 228, 229, 230]) and for isometric operators in Pontryagin spaces see [80]; cf. Chapter 6 for concrete λ-dependent boundary conditions.

The perturbation problems in Section 2.8 and Kre˘ın's formula are closely related. The Sp-perturbation result in Theorem 2.8.3 appears in a similar form in [245]. Standard (additive) perturbations of an unbounded self-adjoint operator yield an analogous situation, where the symmetric operator is maximally nondensely defined [405].

Rigged Hilbert spaces offer a framework where extension theory of unbounded symmetric operators can be developed in a somewhat analogous manner as in the bounded case; see [371, 400, 401, 746]. A closely related approach to extensions of symmetric relations relies on the concept of graph perturbations studied in [203, 204, 205, 206, 238, 264, 267, 270, 399, 403]. There has been a considerable interest in the related concept of singular perturbations, where perturbation elements belong to rigged Hilbert spaces with negative indices; see [10], which contains an extensive list of earlier literature, and see also the notes to Chapter 6 and Chapter 8 in the context of differential operators with singular potentials. General operator models for highly singular perturbations involve lifting of operators in Pontryagin spaces [117, 239, 240, 241, 252, 256, 640, 641, 707].

The interest in boundary triplets and Weyl functions has substantially grown in the last decade, so a complete list of references is beyond the scope of these notes. However, for a selection of papers in which boundary triplet techniques were applied to differential operators and other related problems we refer to [28, 50, 86, 87, 97, 112, 113, 123, 143, 144, 155, 169, 170, 173, 174, 184, 185, 241, 288, 307, 342, 343, 344, 377, 461, 475, 477, 478, 483, 511, 529, 551, 562, 563, 564, 565, 589, 590, 591, 626, 627, 642, 644, 677, 750, 751]. For boundary triplets and similar techniques in the analysis of quantum graphs see [110, 134, 172, 183, 196, 287, 289, 294, 536, 625, 637, 645, 646].

Boundary triplets and their Weyl functions for symmetric operators have been further generalized in [103, 105, 109, 110, 115, 233, 236, 237, 246] by relaxing some of the conditions in the definition of a boundary triplet. Moreover, in the setting of dual pairs of operators (see [525]) boundary triplets have been introduced in [559, 560] and applied, e.g., in [170, 173, 300, 302, 381]; they were specialized to the case of isometric operators in [561]. Boundary triplets and their extensions also occur naturally in a system-theoretic environment, where the underlying operator is often isometric, contractive, or skew-symmetric; cf. [65, 66, 67, 102, 394, 750, 751].

By applying a transform of the boundary triplet, resulting in a Cayley transform of the Weyl function, one gets a connection with the notion of characteristic function, which was used to elaborate an alternative analytic tool for studying extension theory for symmetric operators, for some literature in this direction, see, e.g., [165, 166, 246, 346, 468, 509, 603, 728].

#### **Notes on Chapter 3**

Most of the material in Section 3.1 and Section 3.2 is working knowledge from measure theory. Typical classical textbooks providing a more detailed exposition on Borel measures are [272, 335, 416, 676, 682] and the papers [61, 60, 271] are listed here for symmetric derivatives and the limit behavior of the Borel transform (see Appendix A). We also recommend the more recent monograph [738]. Section 3.3 is a brief exposition of the notions of absolutely continuous and singular spectrum of self-adjoint operators (see [649, 691, 738, 757]) in the context of self-adjoint relations; cf. Section 1.5 and Theorem 1.5.1.

Simplicity of symmetric operators and the decomposition of a symmetric operator into a self-adjoint and a simple symmetric part in Theorem 3.4.4 go back to [496]; cf. [523] for a version of this result for symmetric relations. The local version of simplicity in Definition 3.4.9 appears in papers of Jonas (see, e.g., [437]) and was more recently considered in [119].

The characterization of the spectrum via the limit properties of the Weyl function in Section 3.5 and Section 3.6 is well known for the special case of singular Sturm–Liouville differential operators and the Titchmarsh–Weyl m-function; cf. Chapter 6 and the classics [209, 740, 741] and, e.g., [60, 97, 175, 224, 321, 328, 335, 333, 334, 426, 479, 711, 712] and also [195]. This description was extended in [155] to the abstract setting of boundary triplets and Weyl functions in the case where the underlying symmetric operator is densely defined and simple (on R). These ideas where further generalized in [119, 120] and applied to elliptic partial differential operators. The present treatment in the context of linear relations seems to be new; cf. [265] for Theorem 3.6.1 (i).

Our presentation of the material in Section 3.7 is strongly inspired by the contribution [523] of Langer and Textorius. Due to the limit properties in this section there are important connections between analytic properties of the Weyl functions and the geometric properties of the associated self-adjoint extensions. The characterizations stated in Proposition 3.7.1 and Proposition 3.7.4 have been established initially in [371, 374, 379], where they were used to introduce the notions of generalized Friedrichs and Kre˘ın–von Neumann extensions for nonsemibounded symmetric operators or relations; see the notes on Chapter 5.

Finally, the translation of the results in Section 3.5 and Section 3.6 to arbitrary self-adjoint extensions in Section 3.8 based on the corresponding transformation of the Weyl function (see Section 2.5) is immediate.

#### **Notes on Chapter 4**

One of the central themes in this chapter is the construction of a Hilbert space via a nonnegative reproducing kernel; such a construction has been studied intensively after Aronszjan [59]. For some introductory texts on this topic we refer the reader to [32, 272, 277, 278, 680, 737]. It should be mentioned that there are different approaches avoiding the mechanism of reproducing kernel Hilbert spaces; see for instance Brodski˘ı [165] and for another method Sz.-Nagy and Kor´anyi [604] (sometimes called the ε-method).

Section 4.1 gives a short review of all the facts about reproducing kernel Hilbert spaces that are needed in the monograph. We remind the reader that special reproducing kernels were considered by de Branges and Rovnyak for the setting of Nevanlinna functions or Schur functions [146, 147, 148, 149, 150, 151, 152, 153]; see also [24, 25, 33].

In Section 4.2 the case of operator-valued Nevanlinna functions is taken up first. Theorem 4.2.1 states that the Nevanlinna kernel of a Nevanlinna function M is nonnegative; the proof can be found in [450]. In this case the reproducing kernel approach leads to the realization of the function −(M(λ)+λ)−1 in terms of compressed resolvents in Theorem 4.2.2; see [231, Theorem 2.5 and Remark 2.6] and also [409]. The unitary equivalence of different models in Theorem 4.2.3 is based on a standard argument. If the Nevanlinna function is uniformly strict, then it can be regarded as a Weyl function; see Theorem 4.2.4 as follows by an inspection of the proof of Theorem 4.2.2. The uniqueness of the model in Theorem 4.2.6 is obtained by a further specialization of the proof of Theorem 4.2.3.

Section 4.3 presents a reproducing kernel approach for scalar Nevanlinna functions via their integral representation (see Appendix A) as can be found in [246]; cf. [19, 379, 530]. In the present monograph only the case of scalar Nevanlinna functions is treated in this way; the matrix-valued case is a straightforward generalization. For a rigorous treatment involving operator-valued Nevanlinna functions one should apply [558].

Parallel to the above treatment of operator-valued Nevanlinna functions the case of Nevanlinna families or Nevanlinna pairs is taken up in Section 4.4. The nonnegativity of the Nevanlinna kernel for a Nevanlinna family or Nevanlinna pair in Theorem 4.4.1 is proved via a simple reduction argument. The representation model for Nevanlinna pairs in Theorem 4.4.2 is again given along the lines of Derkach's work [231]; see also [265]. In fact, the proof can be carried forward to show that any Nevanlinna family is the Weyl family of a boundary relation (see the notes for Chapter 2). Another reproducing kernel approach was followed in [95, Theorem 6.1] by a reduction to the corresponding case of Schur functions; cf. [152, 153]. Closely connected is the notion of generalized resolvents given in Definition 4.4.5. It was formalized by McKelvey [572], see also [207]. Its representation in Corollary 4.4.7 is equivalent to the representation of Nevanlinna pairs in Theorem 4.4.2. In Corollary 4.4.9 we present Na˘ımark's dilation result [606, 607] as a straightforward consequence of the characterization of generalized resolvents in Theorem 4.4.8; cf. [659].

Our construction of the exit space extension in Theorem 4.5.2 uses the above representation of generalized resolvents; cf. [497]. The identity in (4.5.3) gives the precise connection between the exit space and the Nevanlinna pair leading to the exit space. Closely related is an interpretation via the coupling method in Section 4.6. By taking an "orthogonal sum of two boundary triplets" as in Proposition 4.6.1 one obtains an exit space extension for one of the original symmetric relations whose compressed resolvent is described in Corollary 4.6.3. This leads to an interpretation of the Kre˘ın formula when the parameter family coincides with a uniformly strict Nevanlinna function; see [373, 375, 376, 630]. However, a similar interpretation exists when the parameter family is a general Nevanlinna family, once it is interpreted as the Weyl family of a boundary relation; see [236, 237, 238]. It has already been explained in the notes on Chapter 2 that it is quite natural to have exit spaces which may be indefinite, in which case the Nevanlinna family must be replaced by a more general object.

And, indeed, more general reproducing kernel spaces can be constructed. For the Pontryagin space situation and reproducing kernels with finitely many negative squares one may consult the monograph [23] and the papers [89, 232, 409, 497, 499, 500]; see also [454, 455, 456, 457, 458, 459]. For the setting of almost Pontryagin spaces, see [452, 775, 777]. Finally, for Kre˘ın spaces we refer to [15, 16, 17, 262, 434, 435, 437, 778].

#### **Notes on Chapter 5**

The approach to semibounded forms in Section 5.1 follows the lines of Kato's work [462], where densely defined semibounded and sectorial forms are treated; see also [2, 49, 246, 346, 359, 552, 649, 650, 690, 691]. An arbitrary form is not necessarily closable, but there is a Lebesgue decomposition into the sum of a closable form and a singular form; see [395, 397, 404, 476, 710]. Versions of the representation theorems for nondensely defined closed forms appear in [14, 44, 393, 661, 710]; see Theorem 5.1.18 and Theorem 5.1.23. Nondensely defined forms appear, for instance, when treating sums of densely defined forms, leading to a representation problem for the form sum; see [295, 296, 389, 390, 392]. The ordering of semibounded forms and semibounded self-adjoint relations in Section 5.2 results in Proposition 5.2.7; the present simple treatment via the Douglas lemma extends [390]; see Proposition 1.5.11. The characterization in terms of the corresponding resolvent operators goes back to Rellich [654]; see also [210, 390, 412, 462]. For the monotonicity principle (Theorem 5.2.11) see [96], and for similar statements [83, 245, 246, 618, 661, 709, 710, 753].

The main properties of the Friedrichs extension, namely Theorem 5.3.3 and Proposition 5.3.6, follow from the present approach via forms; cf. [202, 370, 734]. The square root decomposition approach involving resolvents to describe those semibounded self-adjoint extensions which are transversal with the Friedrichs extension extends [391] (for the nonnegative case). The transversality criterion in Theorem 5.3.8 is an extension of Malamud's earlier result [553]. The introduction of Kre˘ın type extensions in Definition 5.4.2 is inspired by [210]. The minimality of the Kre˘ın type extension is inherited from the maximality of the Friedrichs extension and this yields a characterization of all semibounded extensions whose lower bounds belong to a certain prescribed interval in Theorem 5.4.6. This is the semibounded analog of a characterization of nonnegative self-adjoint extensions in the nonnegative case [210]. The interpretation of the Friedrichs and Kre˘ın type extensions as strong resolvent limits in Theorem 5.4.10 goes back in the operator case to [34] and in the general case to [383]. The present treatment is based on the general monotonicity principle in Theorem 5.2.11. The concept of a positively closable operator (see (iii) of Corollary 5.4.8) was introduced in [34] to detect the nonnegative self-adjoint operator extensions. This notion is closely connected with [490], where self-adjoint operator extensions were determined. Closely related to the Friedrichs and the Kre˘ın–von Neumann extensions is the study of all extremal extensions of a nonnegative or a sectorial operator in [43, 44, 45, 46, 47, 53, 55, 383, 389, 391, 392, 393, 510, 648, 702, 703].

If a symmetric relation is semibounded, then a boundary triplet can be chosen so that A0 is the Friedrichs extension and A1 is a semibounded self-adjoint extension (for instance, a Kre˘ın type extension, when transversality is satisfied). This

type of boundary triplet serves as an extension of the notions of positive boundary triplets introduced in [43, 467]. The main part of the results in Section 5.5 when specialized to the nonnegative case, reduce to the results in [245, 246]. A result similar to Proposition 5.5.8 (ii) can be found in [349, 355], see also the more recent contribution [362] that deals with elliptic partial differential operators on exterior domains. We remark that the sufficient condition in Lemma 5.5.7 is also necessary for the extension AΘ to be semibounded for every semibounded self-adjoint relation Θ in G; see [245, 246]. For an example where Θ is a nonzero bounded operator with arbitrary small operator norm Θ , while AΘ is not semibounded from below, see [378]. In the context of elliptic partial differential operators on bounded domains the boundary triplet in Example 5.5.13 appears implicitly already in Grubb [352]; see also [169, 359, 557] and the notes to Chapter 8 for more details.

The first Green identity appearing in Theorem 5.5.14 establishes a link with the notion of boundary pairs for semibounded operators and the corresponding semibounded forms in Section 5.6. Definition 5.6.1 of a boundary pair for a semibounded relation S is adapted from Arlinski˘ı [44], see also [50] for the context of generalized boundary triplets. This notion makes it possible to describe the closed forms associated with all semibounded self-adjoint extensions of a given semibounded relation S by means of closed semibounded forms in the parameter space. In particular, Theorem 5.6.11 contains the description of all nonnegative closed forms generated by the nonnegative self-adjoint extensions in [53]; for a similar result in the case of maximal sectorial extensions see [44]. By connecting boundary pairs with compatible boundary triplets (see Theorem 5.6.6 and Theorem 5.6.10) one obtains an explicit description of the forms associated with all semibounded self-adjoint extensions by means of the boundary conditions, see Theorem 5.6.13.

We remind the reader of the study of nonnegative self-adjoint extensions of a densely defined nonnegative operator as initiated by von Neumann [610]; see [280, 309, 310, 724, 773] and the papers by Kre˘ın [491, 492, 493, 494]. Kre˘ın's analysis was complemented by Vishik [747] and Birman [139]; see also [2, 14, 58, 157, 217, 295, 346, 347, 352, 354, 372, 464, 714]. For an operator matrix completion approach see [383, 472], see also [81] for an extension in a Pontryagin space setting. This approach has its origin in the famous paper of Shmuljan [704]. The notion of boundary pairs for forms can also be traced back to Kre˘ın [493, 494], Vishik [747], and Birman [139]. Some further developments can be found, e.g., in [36, 43, 44, 47, 48, 54, 56, 57, 248, 301, 357, 359, 363, 553, 555, 556], where also accretive and sectorial extensions, and associated closed forms are treated, and see also [552, 580, 581, 582, 636, 725]. A related concept of boundary pairs designed for elliptic partial differential operators and corresponding quadratic forms was recently proposed by Post [647].

A study of symmetric forms which are not semibounded and concepts of generalized Friedrichs and Kre˘ın–von Neumann extensions goes beyond the scope of the present text. However, an interested reader may consult in this direction, e.g., [125, 215, 216, 303, 306, 364, 371, 374, 379, 570, 571].

#### **Notes on Chapter 6**

The main textbooks in this area are [2, 26, 72, 208, 281, 284, 414, 420, 438, 489, 542, 574, 608, 663, 724, 740, 741, 754, 757, 782]. The Sturm–Liouville differential expressions that we consider here have coefficients which are assumed to satisfy some weak integrability conditions giving rise to quasiderivatives (where under stronger smoothness conditions ordinary derivatives would suffice); see for instance [724] where already such quasiderivatives were used. Most of the material concerning singular Sturm–Liouville equations has been stimulated by the work of Weyl [758, 759, 760]; see also [761]. For later work on operators of higher order see, for instance, [200, 201, 465, 469, 470].

Section 6.1 and Section 6.2 contain standard material, where we follow parts of [414, 757]; for the quasiderivatives in the limit-circle case we refer to [312]. The treatment of the regular and limit-circle cases in Section 6.3 is straightforward; see [312] and also [12] for a boundary triplet in the limit-circle case. For an alternative description of boundary conditions and Weyl functions in the regular case we mention the recent papers [197, 199, 332] using the concept of boundary data maps. The limit-point case is treated in Section 6.4. In the proof of Proposition 6.4.4 concerning the simplicity of the minimal operator we follow the arguments of R.C. Gilbert [336]; cf. [263]. The treatment of the Fourier transform in Lemma 6.4.6 and Theorem 6.4.7 uses the theory in Appendix B. The surjectivity is direct in this case. The statement in Lemma 6.4.8 that the Fourier transform provides a model for the Weyl function uses an argument provided in [281].

In general, interface conditions are a tool to paste together several boundary value problems. The interface conditions which we consider in Section 6.5 make it possible in the case of two endpoints in the limit-point case to consider minimal operators [339, 740] and to apply the coupling of boundary triplets as explained in Section 4.6. This automatically leads to some kind of exit space extensions. As this goes beyond the present text, we only refer to [336, 338, 441, 713].

The present theory of subordinate solutions in Section 6.7 goes back to D.J. Gilbert and D.B. Pearson [335]; see also [333, 334, 463]. Further contributions can be found in [433, 533, 655, 656, 657]; see also [738]. Our presentation is modelled on [388].

If the minimal operator is semibounded one can apply the form methods from Chapter 5. In the regular case the treatment of the forms associated with the Sturm–Liouville expression in Section 6.8 is based on the inequality in Lemma 6.8.2 and Theorem 5.1.16; see [212, 493, 494, 757]. This makes it possible to introduce boundary pairs which are compatible with the given boundary triplet. Section 6.9 and Section 6.10 contain preparatory material so that boundary pairs can be introduced also when the endpoints are singular. Section 6.9 on Dirichlet forms is inspired by Rellich [654] and Kalf [448]. The proof of Lemma 6.9.7 seems to be new. Section 6.10 is concerned with principal and nonprincipal solutions; cf. [198, 368, 369, 535] and, in particular, [616]. The Hardy type inequalities in Lemma 6.10.1 go back to [449]. The treatment is in some sense folklore, but Theorem 6.10.9 seems to be new. Our handling of the regular case in Section 6.8 serves as model for the semibounded singular cases in Section 6.11 and Section 6.12. These two sections are influenced by Kalf [448]; see also [671, 672] and a recent contribution [168]. The determination of the Friedrichs extension in our treatment goes hand in hand with the boundary pair; see also [310, 311, 508, 597, 600, 601, 615, 616, 654, 661, 671, 672, 779, 782].

The case of an integrable potential on a half-line is treated in Section 6.13. It appears already in [740], where the corresponding spectral measure is determined. The construction of solutions with a given asymptotic behavior can be found for instance in [542, 740]. The example with the P¨oschl–Teller potential at the very end of Section 6.13 can be found in, e.g., [2].

There is a large amount of literature devoted to special topics. For λ-dependent boundary conditions we just refer to [117, 313, 314, 675, 687, 688, 689, 749] and [261, 263] for Pontryagin exit spaces. The determination of the Kre˘ın–von Neumann extension and other nonnegative extensions can be found in [212, 350]. For singular perturbations associated with Sturm–Liouville operators, see for instance [331] and the later papers [9, 28, 29, 124, 171, 182, 282, 315, 316, 479, 480, 481, 482, 507, 531, 532, 549, 550]; for δ-point interactions we refer to [8, 293]. Special properties of the Titchmarsh–Weyl coefficient have been studied in many papers; we just mention [130, 292, 366, 367]. Already early in the 20th century there was an interest in boundary value problems with conditions at interior points and, more general, integral boundary conditions; for instance, see [135, 583] and for a later review [762]. It was pointed out by Coddington [203, 204] that such conditions can be described in tems of relation extensions of a restriction of the usual minimal operator; see for a brief selection also [188, 189, 190, 203, 204, 205, 206, 238, 264, 267, 270, 399, 403, 484, 485, 486, 697, 784].

The topic of semibounded self-adjoint extensions in Section 5.5 and Section 5.6 is closely related to problems that are called left-definite in the literature; we only mention [13, 131, 473, 474, 488, 545, 698, 699, 708, 748] and the references therein. Another case of interest is when the weight function in the Sturm–Liouville equation changes sign, in which case Kre˘ın spaces come up naturally: the following is just a limited selection [116, 122, 123, 138, 223, 225, 304, 305]. Under certain circumstances the Weyl function in the limit-point case (of a usual Sturm–Liouville operator) belongs to the Kac class; see for instance [420]. For further results in this direction, see [384, 385, 386, 387]; in these cases there is a distinguished selfadjoint extension, namely the generalized Friedrichs extension; see the notes on Chapter 5. Sturm–Liouville equations with vector-valued coefficients are beyond the scope of this text; see [345, 346] and for a recent contribution [330].

#### **Notes on Chapter 7**

Canonical systems of differential equations are discussed in the monographs [72, 340, 653, 681]. The spectral theory for such systems has been developed in varying degrees of generality in an abundance of papers [62, 63, 64, 85, 161, 162, 163, 213, 263, 265, 279, 422, 423, 424, 425, 427, 428, 487, 575, 592, 612, 613, 614, 620, 667, 668, 669, 687, 688, 689, 692, 693, 694, 695, 696]. In [520] there is a general procedure to reduce systems to a canonical form. Many investigations concerning boundary problems can be written in terms of canonical systems; see for a general procedure [619, 693], so that for instance [129, 211] can be subsumed. As a particular example we mention the system of second-order differential equations studied in the dissertation of a student of Titchmarsh [191, 192, 193, 194]. Frequently conditions are imposed on the canonical system so that there is a full analogy with the usual operator treatment. However, in general one is confronted with relations.

The first treatment of canonical systems in terms of relations is due to Orcutt [619]; see also [97, 408, 471, 524, 538] and [11, 593, 594]. In our opinion the present case of 2×2 systems already serves as a good illustration of the various phenomena that may occur. The interest in 2×2 systems is justified by the work of de Branges [146, 147, 148, 149]; see also [278, 504, 528, 715, 716, 766, 767, 768, 769, 770, 771, 772]. Via these systems one may also approach Sturm–Liouville equations with distributional coefficients as, for instance, in [281, 283]. In Remling [658] and the forthcoming book [666] by Romanov our systems are treated with the de Branges results in mind. For a number of topics we rely on the lecture notes by Kaltenb¨ack and Woracek [460].

Sections 7.1, 7.2, and 7.3 contain preparatory material. The inequalites (7.1.1) and (7.1.3) are standard for Bochner integrals. The construction of the Hilbert space L2 Δ(ı) follows the treatment in [460]; see also [276, 670]. For further information concerning these and more general spaces, see [440] and the expositions in [2, 276]. The treatment of the square-integrable solutions in Section 7.4 is based on the monotonicity principle in Theorem 5.2.11; cf. [97]. For a different treatment, see for instance [612]. Section 7.5 is devoted to definite systems. The present notion of definiteness can be found in [340, 619]; in the literature sometimes a more restrictive form of definiteness is used. Proposition 7.5.4 can be found in [614, Hilfsatz (3.1)] and [471]; for a more abstract treatment, see [98]. The modification of solutions is avoided in [619]; the present argument in Proposition 7.5.6 seems to be new. The minimal and maximal (multivalued) relations associated with canonical systems can be found in Section 7.6. They were originally introduced by Orcutt [619]; see also Kac [442, 443, 444, 445] and [408]. The extension theory for them naturally involves (multivalued) relations. In [97] it is indicated how all such systems fit in the boundary triplet scheme; see also [471, 538].

In Section 7.7 it is assumed that the endpoints are (quasi)regular, in which case a boundary triplet is constructed in Theorem 7.7.2. The resolvent of the self-adjoint extension ker Γ0 is an integral operator whose kernel belongs to the Hilbert–Schmidt class. In this way we can show that the operator part of the minimal relation is simple. In Section 7.8 it is assumed that one of the endpoints is in the limit-point case. We prove the simplicity of the operator part of the minimal relation along the lines of [337], cf. [263] (see also Chapter 6 for the Sturm– Liouville case). The treatment of the Fourier transform in the limit-point case uses

Appendix B and parallels the treatment in Chapter 6. Subordinate solutions for canonical systems are introduced in Section 7.9; cf. [388]. Here again we follow the results for the Sturm–Liouville case in Section 6.7; see also the corresponding notes in Chapter 5, where the appropriate references can be found.

The discussion of the special cases in Section 7.10 is just an indication of the possibilities; we could also pay attention to, for instance, the so-called Dirac systems. The connection with the Sturm–Liouville case and its generalization in [281] is only briefly indicated. The special case in Theorem 7.10.1 is modelled on [460]. For the connection of canonical systems with strings see, for instance, [453]. Finally, we would like to mention that the approach via boundary triplets allows also λ-dependent boundary conditions; for some related papers we refer to [221, 263, 265, 675, 687, 688, 689, 742].

#### **Notes on Chapter 8**

The notion of Gelfand triples or riggings of Hilbert spaces in Section 8.1 is often used in the treatment of partial differential equations and Sobolev spaces, and can be found in a similar form in Berezanski˘ı's monograph [133] (see also the textbook [774]); cf. [132] and the contributions [534, 537] by Lax and Leray. There are many well-known textbooks on Sobolev spaces, among which we mention here only the monographs [3, 5, 164, 136, 284, 291, 351, 569, 584, 744, 783]. In our opinion a very useful source for trace maps, the Green identities, and similar related results (also for nonsmooth domains) is the monograph [573] by McLean, see also the list of references therein. For the description of the spaces Hs(∂Ω) in Corollary 8.2.2 as domains of powers of the Laplace–Beltrami operator on ∂Ω see [322, 544, 567]. In some cases it is also convenient to use powers of a Dirichlet-to-Neumann map, as in [114].

The discussion of the minimal and maximal operators in Section 8.3 is standard, e.g., Proposition 8.3.1 can be found in Triebel's textbook [745]. The H2 regularity in Theorem 8.3.4 for smooth domains can be found in [167] and [84], see also [4, 308, 544, 517] or [1, Theorem 7.2] for a recent very general result on the H2-regularity of the Neumann operator (and more general) on certain classes of nonsmooth bounded and unbounded domains. As explained in Section 8.7, for the class of Lipschitz domains the H2-regularity of the Dirichlet and Neumann Laplacian up to the boundary fails and has to be replaced by the weaker H3/2 regularity; cf. [431, 432] and [92, 115, 323, 324, 325, 326] for some recent closely related works dealing with Schr¨odinger operators on Lipschitz domains. In this context we also mention the papers [585, 586, 587, 588] by Mitrea and Taylor for general layer potential methods in Lipschitz domains on Riemannian manifolds. It is worth mentioning that the resolvent of the Neumann Laplacian in Proposition 8.3.3 is not necessarily compact if the boundary of the bounded domain Ω is not of class C2 (or not Lipschitz), see [415] for a well-known counterexample using a rooms-and-passages domain. For similar unusual spectral properties we also mention Example 8.4.9 on the essential spectrum of self-adjoint realizations of the

Laplacian on a bounded domain, which however is very different from a technical point of view. In a related context the existence of self-adjoint extensions with prescribed point spectrum, absolutely continuous, and singular continuous spectrum in spectral gaps of a fixed underlying symmetric operator was also discussed in [6, 7, 154, 156, 158, 159, 160]. For our purposes Theorem 8.3.9 and Theorem 8.3.10 play an important role; for the case of C∞-smooth domains such results can be found in [544], see also [352, 354]. The present versions of the extension theorems are inspired by slightly different considerations in [115]; the variant for Lipschitz domains in Theorem 8.7.5 can be proved by means of a similar technique (see also [92] for a comprehensive discussion). We remark that the definition and topologies on the spaces G0 and G1 in (8.7.4)–(8.7.5) from [92, 115] are partly inspired by abstract considerations in [246] for generalized boundary triplets. Concerning Section 8.3, as a final comment we mention that for more general second-order elliptic operators in bounded and unbounded domains Proposition 8.3.13 can be found in [118, 119, 120].

The boundary triplet in Theorem 8.4.1 can also be found in, e.g., [105, 169, 173, 359, 557, 643] and extends with some simple modifications to second-order and 2m-th order elliptic operators with variable coefficients. In a different form this boundary triplet is already essentially contained in the well-known work of Grubb [352], where all closed extensions of a minimal operator elliptic partial differential operator were characterised by nonlocal boundary conditions; see also the early contribution [747] by Vishik and the fundamental paper [140] by Birman. In fact, it seems that the more recent paper [27] by Amrein and Pearson on a generalisation of Weyl–Titchmarsh theory for Schr¨odinger operators inspired many operator theorists to investigate partial differential operators from an extension theory point of view. Various papers based on boundary triplet techniques and related methods were published in the last decade; besides those papers listed before we mention as a selection here [18, 90, 91, 103, 107, 108, 251, 323, 324, 360, 429, 566, 568, 635, 647, 678] in which, e.g., Dirichlet-to-Neumann maps and Kre˘ın type resolvent formulas are treated. For self-adjoint realizations of the Laplacians, Schr¨odinger operators, and more general second-order elliptic differential operators in nonsmooth domains we refer to, e.g., [1, 92, 115, 326, 358]. Particular attention has been paid to the spectral properties of realizations with local and nonlocal Robin boundary conditions in [41, 106, 179, 180, 226, 325, 361, 413, 543, 628, 629, 665]. Furthermore, the recent contributions [35, 36, 37, 38, 39, 40] by Arendt, ter Elst, and coauthors form an interesting series of papers on elliptic differential operators and Dirichlet-to-Neumann operators based mainly on form methods.

The present treatment of semibounded Schr¨odinger operator in Section 8.5 with the help of boundary pairs and boundary triplets seems to be new. However, the problem of lower boundedness was also discussed with different methods in [349, 355, 362]. The Kre˘ın–von Neumann extension which is of special interest in this context was investigated in, e.g., [14, 68, 69, 70, 71, 93, 181, 356, 362, 578, 579]. For coupling methods for elliptic differential operators based on boundary triplet

techniques in the spirit of Section 8.6 we refer to the recent paper [88], where also an abstract version of the third Green identity was proved. Finally, we mention that various classes of λ-dependent boundary value problems for elliptic operators can be treated in such a context, see, e.g., [87, 103, 137, 285, 286].

#### **Notes on Appendices A–D**

The appendices were included for the convenience of the reader. Here we collect some notes for each of the appendices A–D.

In Appendix A we present the basic properties of Nevanlinna functions that are needed in the text. The Stieltjes inversion formula for the Borel transform is folklore. The integral representation of scalar Nevanlinna functions is derived in a classical way involving the Helly and Helly–Bray theorems; see [72, 763]. The inversion formula in Lemma A.2.7 is standard; see [272, 446]. For the present purposes it suffices to approach the integration with respect to operator-valued measures in terms of improper Riemann integrals. For the operator measure version see for instance [691]. For Kac functions, Stieltjes functions, and inverse Stieltjes functions see [49, 52, 329, 407, 446, 447, 505]. As far as we know, the proof of Proposition A.5.4 is new; cf. [572]. For related work see [318, 319, 329, 602]. The case of generalized Nevanlinna functions was initiated in the works of Kre˘ın and Langer [499, 500]; see [145, 225, 232, 239, 242, 255, 380, 411, 434, 435, 437, 546, 547, 548] for further developments.

For the general notion of Fourier transforms in Appendix B associated with Sturm–Liouville equations, see [539, 540, 542, 700, 701, 740, 780, 781]. Our basic idea here is inspired by the treatment in [208]. The arguments establishing the surjectivity of the Fourier transform are also inspired by [281, 715, 716]. Fourier transforms that are partially isometric go back at least to [204] in a treatment connected with multivalued operators. The discussion in this section has connections with the treatment of Kre˘ın's directing functionals in [495, 518, 526, 527].

Necessary and sufficient conditions (as in Appendix C) for the sum of closed subspaces have a long history. The concept of opening of a pair of subspaces goes back to Friedrichs and Dixmier; see for instance [214, 249, 250]. Lemma C.4 goes back to [246].

The main reference for Appendix D is the paper by Douglas [273]; see also [30, 299]. There have been many generalizations of the Douglas paper. We only mention a particular direction of extension, namely [639, 684]; see also [402]. The application in Proposition 1.5.11 in Chapter 1 has the same flavor, but is obtained by reduction to the case in this appendix; see [390].

## **Bibliography**

## **List of Symbols**

For a linear relation H the following notation is used:

© The Editor(s) (if applicable) and The Author(s) 2020 J. Behrndt et al., *Boundary Value Problems, Weyl Functions, and Differential Operators*, Monographs in Mathematics 108, https://doi.org/10.1007/978-3-030-36714-5

## **Index**

absolutely continuous, 366, 503 absolutely continuous closure, 180 antitonicity, 303 Borel transform, 172, 631 boundary pair, 344 space, 115 triplet, 107 boundary conditions Dirichlet, 390, 593 essential, 360 λ-dependent, 425 natural, 360 Neumann, 390, 593 periodic, 394 Robin, 609 separated, 394 bounded operator, 15 C2-domain, 583 C2-hypograph, 583 canonical system definite, 522 homogeneous, 504 inhomogeneous, 504 real, 509 regular, 510 singular, 510 solution, 504 trace-normed, 573 Cayley transform, 22 compatible, 345 completely non-self-adjoint, 190 completion problem, 130 compressed resolvent, 156 continuous closure, 180 contractive, 16

core, 55, 290, 298 coupling (orthogonal), 274, 414, 616 defect, 25 defect number, 44 dilation theorem, 269 Dirichlet -to-Neumann map, 594, 599 operator, 591 trace operator, 586, 595 disjoint, 68 distribution (regular), 587 Douglas lemma, 701 dual pairing, 578 elliptic regularity, 593 essential closure, 180 exit space, 158 extension, 13, 345 finite-rank perturbation, 163 flip-flop operator, 30 form bounded from below, 282 closable, 286 closed, 285 closure, 287 domain, 282 extension, 282 inclusion, 282 lower bound, 282 nonnegative, 282 quadratic, 282 restriction, 282 semibounded, 283 sesquilinear, 282 sum, 282

symmetric, 282

© The Editor(s) (if applicable) and The Author(s) 2020 J. Behrndt et al., *Boundary Value Problems, Weyl Functions, and Differential Operators*, Monographs in Mathematics 108, https://doi.org/10.1007/978-3-030-36714-5

t-convergence, 284 Fourier transform, 406, 419, 552, 581, 678, 685 Friedrichs extension, 312 functional calculus, 49 fundamental matrix, 507 fundamental system, 370

γ-field, 118 gap, 697 Gelfand triple, 578 generalized resolvent, 266 graph, 12 Green identity, 369 abstract, 108 abstract first, 341 first, 444, 586 second, 587, 598 Gronwall's lemma, 487

Hilbert–Schmidt operator, 537

indefinite inner product, 75 integrable potential, 483 interface condition, 412 intermediate extension, 68 inverse Cayley transform, 22 inverse Stieltjes function, 669 isometric, 16, 76

Kac function, 663 kernel holomorphic, 224 nonnegative, 224 symmetric, 224 uniformly bounded, 224 KLMN theorem, 290 Kre˘ın -Na˘ımark formula, 159 -von Neumann extension, 321 formula, 148 space, 75 type extension, 321

Lagrange identity, 108, 369, 505 Lebesgue decomposition, 39, 170 limit-circle, 374, 518 limit-point, 374, 518

Lipschitz domain, 625 Lipschitz hypograph, 625 M¨obius transform, 20, 78 maximal operator, 380, 589 maximal relation, 525 measure absolutely continuous, 170 Borel, 169 derivative, 170 finite, 170 growth point, 178 minimal support, 171 operator-valued, 652 pure point, 170 regular, 170 singular, 170 singular continuous, 170 support, 171 symmetric derivative, 173 minimal operator, 381, 589 minimal relation, 526 model (minimal), 237 monotonicity principle, 307, 310 Moore–Penrose inverse, 41, 699 Neumann operator, 591 Neumann trace operator, 586, 598 Nevanlinna family, 100 function, 638, 655 kernel, 235, 261 pair, 102 nonoscillatory, 442 nonprincipal solution, 459 opening, 695 operator, 12 -valued integral, 645 part, 40 range, 88 orthogonal companion, 75 oscillatory, 442 parameter space, 115 parametric representation, 87 Pl¨ucker identity, 368

#### Index 771

Poincar´e inequality, 585 point of regular type, 23 preminimal operator, 381, 589 preminimal relation, 526 principal solution, 459 Q-function, 139 quasi-derivative, 385 quasiregular endpoint, 510 regular endpoint, 367, 510 regular part, 39 relation accumulative (maximal), 58 adjoint relation, 30, 74 bounded from below, 44 closable operator, 16 closed, 16 closure, 16 componentwise sum, 12 dissipative (maximal), 58 domain, 12 inverse, 12 kernel, 12 lower bound, 44 multivalued part, 12 nonnegative, 44 ordering, 301 orthogonal sum, 15 product, 14 range, 12 self-adjoint, 42 semibounded, 44 square root, 54 sum, 14 symmetric (maximal), 42 representation theorem, 293, 299 reproducing kernel Hilbert space, 224 reproducing kernel property, 225 resolvent identity, 17, 96 operator, 17, 96 relation, 17 set, 23 restricted boundary triplet, 142 restriction, 13 rigged Hilbert space, 578

Schatten–von Neumann ideal, 165 Schr¨odinger operator, 588, 616 self-adjoint extension, 43 simple, 190 simple with respect to Δ, 193 singular continuous subspace, 185 endpoint, 368, 510 part, 39 value, 165 Sobolev space, 581 solution matrix, 506 spectral function, 49 measure, 49 projection, 49 spectrum absolutely continuous, 186 continuous, 23 discrete, 48 essential, 48 point, 23 residual, 23 singular, 186 singular continuous, 186 square-integrable with respect to Δ, 501 Stieltjes inversion formula, 631, 641 Stone's formula, 50, 634 Straus family, ˇ 156 strong graph convergence, 80 graph limit, 80 resolvent convergence, 81 resolvent limit, 81 Sturm–Liouville operator, 366 subordinate solution, 425, 560 subspace absolutely continuous, 185 continuous, 186 defect, 46 hypermaximal neutral, 75 invariant, 188 neutral, 75 nonnegative, 75 nonpositive, 75

pure point, 185 singular, 186

tight, 88 trace map, 586 trace operator, 586 transformation, 134 transversal, 68

uniformly strict, 663 unitarily equivalent, 33 , 145 unitary, 33 , 76

variation of constants, 370 , 507 von Neumann's first formula, 71 von Neumann's second formula, 71 , 73

Weyl function, 121 Weyl's alternative, 378 , 518 Wronskian determinant, 368