# Operators which preserve a positive definite inner product

E. Andruchow111Instituto Argentino de Matemática, ‘Alberto P. Calderón’,
CONICET, Saavedra 15 3er. piso, (1083) Buenos Aires, Argentina  and
Universidad Nacional de General Sarmiento, J.M. Gutierrez 1150, (1613) Los
Polvorines, Argentina e-mail<EMAIL_ADDRESS>
###### Abstract

Let ${\cal H}$ be a Hilbert space, $A$ a positive definite operator in ${\cal
H}$ and $\langle f,g\rangle_{A}=\langle Af,g\rangle$, $f,g\in{\cal H}$, the
$A$-inner product. This paper studies the geometry of the set

${\cal I}_{A}^{a}:=\\{\hbox{ adjointable isometries for }\langle\ ,\
\rangle_{A}\\}.$

It is proved that ${\cal I}_{A}^{a}$ is a submanifold of the Banach algebra of
adjointable operators, and a homogeneous space of the group of invertible
operators in ${\cal H}$, which are unitaries for the $A$-inner product. Smooth
curves in ${\cal I}_{A}^{a}$ with given initial conditions, which are minimal
for the metric induced by $\langle\ ,\ \rangle_{A}$, are presented. This
result depends on an adaptation of M.G. Krein’s extension method of symmetric
contractions, in order that it works also for symmetrizable transformations
(i.e., operators which are selfadjoint for the $A$-inner product).

2020 MSC: 47A05, 47A62, 58B20, 58B10

Keywords: $A$-isometries, $A$-unitaries, compatible subspaces, symmetrizable
transformations.

##  1 Introduction

Let $A$ be a positive contraction in the Hilbert space $({\cal H},\langle\ \
,\ \ \rangle)$ with trivial nullspace $N(A)=\\{0\\}$. Denote by $\langle\ \ ,\
\ \rangle_{A}$ the inner product defined by $A$:

$\langle f,g\rangle_{A}=\langle Af,g\rangle,\ f,g\in{\cal H}.$

This paper considers the operators which preserve this inner product, which
will be called $A$-isometries. That is, a bounded operator $T\in{\cal B}({\cal
H})$ is called an $A$-isometry if

$\langle ATf,Tg\rangle=\langle Af,g\rangle,$

or equivalently, $T^{*}AT=A$. The focus will be on $A$-isometries which
additionally admit an adjoint for the $A$-inner product: there exists
$S\in{\cal B}({\cal H})$ such that $\langle Tf,g\rangle_{A}=\langle
f,Sg\rangle_{A}$, for $f,g\in{\cal H}$. In general, an operator $B$ in ${\cal
H}$ will be called $A$-adjointable (or just adjointable) if it admits an
adjoint for the $A$-inner product, which shall be denoted by $C=A^{\sharp}$.
Denote by

${\cal I}_{A}:=\\{T\in{\cal B}({\cal H}):T\hbox{ is }A-\hbox{isometric}\\}$

and

${\cal I}_{A}^{a}:=\\{T\in{\cal I}_{A}:T\hbox{ is }A-\hbox{adjointable}\\}$
(1)

It will be shown that there are $A$-isometries which are not $A$-adjointable.
It is the latter set ${\cal I}_{A}^{a}$ which admits a differentiable
structure, and which is a homogeneous space of a Banach-Lie group. Moreover,
endowed with a natural Finsler metric (namely, the one induced by he $A$-inner
product), curves which have minimal length for given initial conditions are
computed.

The contents of the paper are the following. In Section 2 the basic facts
needed are stated, on operators in Hilbert spaces with two norms (these facts
are taken from [11], [13], [10]); examples of adjointable and non adjointable
$A$-isometries are presented. Also Theorem 2.6 is proved, characterizing
adjointable $A$-isometries. A result by R. Douglas [9] is used, on existence
of solutions of the operator equation $AX=B$. In Section 3 the Wold
decomposition of $A$-isometries is briefly analized. Section 4 contains
Theorem 4.3, stating that ${\cal I}_{A}^{a}$ is a $C^{\infty}$-submanifold of
the $*$-Banach algebra of adjointable operators (with a suitable norm), and a
homogeneous space of the Banach-Lie group of $A$-unitary operators (see the
definition below). Section 5 presents an adaptation of Krein’s method for the
extension of symmetric transformations with norm constraints, to work in the
context of symmetrizable transformations (Lemma 5.1). We believe that this
section is of independent interest. This result is used in Section 6 to prove
Theorem 6.1, which computes minimal curves in ${\cal I}_{A}^{a}$ satisfying
given initial conditions. Section 7 treats the action of the restricted group
of $A$-unitaries, the orbits of the action of this group are characterized
(Theorem 7.2), and it is proved that these are also $C^{\infty}$-manifolds and
homogeneous spaces (Proposition 7.3).

Let us finish this section with basic facts and notations. Denote by ${\cal
L}$ the completion of the pre-Hilbert space $({\cal H},\langle\ \ ,\ \
\rangle_{A})$. Note that $T\in{\cal I}_{A}$ extends to an isometry ${\bf T}$
acting in ${\cal L}$ (denote the inner product on ${\cal L}$ by $\langle\ \ ,\
\ \rangle_{\cal L}$). Conversely, an isometry ${\bf V}\in{\cal B}({\cal L})$
that leaves the dense subspace ${\cal H}$ invariant, i.e. ${\bf V}({\cal
H})\subset{\cal H}$, induces an operator $V={\bf V}|_{\cal H}$ in ${\cal H}$,
which is an $A$-isometry.

This study relies on the theory of symmetrizable operators, (or more broadly,
operators in Hilbert spaces with two norms), developed initially (and
independently) by M.G. Krein [11] and P.D. Lax [13], and extended afterwards
by I.C. Gohberg and M.K. Zambickii [10].

Upper case letters $T,S,X,G,\dots$ will denote operators acting in ${\cal H}$
(with adjoints in ${\cal H}$ denoted $T^{*},S^{*},X^{*},G^{*},\dots$). Their
eventual extensions to ${\cal L}$ will be denoted with capital bold letters:
${\bf T},\SS,{\bf X},{\bf G},\dots$, and adjoints ${\bf T}^{*},\SS^{*},{\bf
X}^{*},{\bf G}^{*},\dots$ (in ${\cal L}$). That is, which adjoint is refered
to, will depend on the context. Vectors in ${\cal H}$ will be denoted $f,g,h$
and vectors in ${\cal L}$ with Greek letters $\varphi,\gamma,\eta$.

A special class of $A$-isometries, is given by the $A$-unitary operators:
$G\in{\cal B}({\cal H})$ is called $A$-unitary if it is an $A$-isometry which
is invertible in ${\cal H}$. Denote by

${\cal U}_{A}=\\{G\in{\cal B}({\cal H}):G\hbox{ is invertible and
}G^{*}AG=A\\},$

the group of $A$-unitaries. Note that $A$-unitaries are adjointable. In [3] it
was shown that ${\cal U}_{A}$ is a $C^{\infty}$ Banach-Lie group. Clearly,
${\cal U}_{\cal A}$ acts on ${\cal I}_{A}$ and on ${\cal I}_{A}^{a}$ by left
multiplication:

$G\cdot T=GT\in{\cal I}_{A}\ ,\ \hbox{ if }G\in{\cal U}_{A}\hbox{ and
}T\in{\cal I}_{A},$

and clearly $GT$ is adjointable if $T$ is adjointable. Again, $G\in{\cal
U}_{A}$ extends to a unitary operator ${\bf G}$ such that ${\bf G}({\cal
H})={\cal H}$.

##  2 Symmetrizable operators

Denote by $\sigma_{\cal H}(T)$ the spectrum of $T$ as an operator in ${\cal
H}$, and by $\sigma_{\cal L}({\bf T})$ the spectrum of its extension to ${\cal
L}$ (when it exists). We will say that $\lambda$ belongs to the point spectrum
of $T$ if $0<\dim(N(T-\lambda 1))$, and $\lambda$ is said to have finite
multiplicity if this dimension is finite. An operator $B$ acting in ${\cal H}$
will be called $A$-symmetric (or symmetrizable) if it is symmetric for the
$A$-inner product. Let us recall the following facts, adapted from their
original broader context to our case:

######  Theorem 2.1.

(See M.G.Krein [12], P.D. Lax [13]) Let $B$ be an $A$-symmetric operator. The
following assertions hold:

  1. 1.

${\bf B}$ exists and is bounded selfadjoint operator in ${\cal L}$.

  2. 2.

$\sigma_{\cal L}({\bf B})\subset\sigma_{\cal H}(B)$.

  3. 3.

If $\lambda$ belongs to the point spectrum of $B$ as an operator in ${\cal
H}$, then $\lambda$ belongs to the point spectrum of ${\bf B}$ as an operator
in ${\cal L}$. Moreover, if $\lambda$ has finite multiplicity, then the
$\lambda$-eigenspaces over ${\cal H}$ and ${\cal L}$ are the same.

  4. 4.

If $B$ is a compact operator in ${\cal H}$, then ${\bf B}$ is a compact
operator in ${\cal L}$ and $\sigma_{\cal L}({\bf B})=\sigma_{\cal H}(B)$.

######  Remark 2.2.

I.C. Gohberg and M.I. Zambickii [10] extended these results to operators in
Banach spaces with two norms. In particular, in our context, they proved that
if an operator $B$ in ${\cal H}$ is compact and $A$-adjointable (they called
adjointable operators proper), the its extension ${\bf B}$ to ${\cal L}$ is
also compact, with the same point spectrum, and also the same eigenspaces of
finite multiplicity.

Denote by

${\cal B}_{A}({\cal H})=\\{B\in{\cal B}({\cal H}):B\hbox{ is
}A-\hbox{adjointable}\\}$ (2)

If $A$ is not invertible, ${\cal B}_{A}({\cal H})$ is not closed in ${\cal
B}({\cal H})$, neither is the $A$-adjoint map $B\mapsto B^{\sharp}$ a
continuous map (in the norm topology). ${\cal B}_{A}({\cal H})$ is endowed
with he norm

$|B|:=\max\\{\|B\|,\|B^{\sharp}\|\\}.$

Then $({\cal B}_{A}({\cal H}),|\ \ |)$ becomes an involutive Banach algebra.

######  Remark 2.3.

It is not difficult to see that if $B$ is $A$-symmetric, then $\|{\bf
B}\|\leq\|B\|$. Also note that $A$ itself is $A$-symmetric, and that its
extension ${\bf A}$ remains positive definite.

A closed linear subspace ${\cal S}\subset{\cal H}$ is called compatible with
$A$, $A$-compatible, or shortly compatible, if it admits a supplement which is
orthogonal with respect to the inner product defined by $A$. In [4], the
compatible Grassmannian was studied, namely

$Gr_{A}=\\{{\cal S}\subset{\cal H}:{\cal S}\hbox{ is closed and compatible
with }A\\}.$

Since $A$ has trivial nullspace, if $S$ is compatible with $A$, then the
supplement is unique, and it is given by $A({\cal S})^{\perp}$. This allows us
to identify each compatible subspace ${\cal S}$ with the idempotent $Q_{\cal
S}$ with range ${\cal S}$ and nullspace $A({\cal S})^{\perp}$. Thus the
compatible Grassmannian may be regarded as the following set

$Gr_{A}=\\{Q\in{\cal B}({\cal H}):Q^{2}=Q,Q^{*}A=AQ\\}$ $=\\{Q\in{\cal
B}({\cal H}):Q^{2}=Q\hbox{ is adjointable and symmetrizable}\\}.$

In other words, ${\bf Q}=P_{\overline{R(Q)}}$. Consider in $Gr_{A}$ the
topology inherited from the norm of ${\cal B}_{A}({\cal H})$. The proof of the
afore-mentioned facts and examples of compatible and non-compatible subspaces
can be found in [6], [7], where a systematic study of compatible subspaces was
done. The notion of compatible subspaces goes back to A. Sard [17], who
introduced an equivalent definition under a different terminology, to give an
operator theoretic approach to problems in approximation theory (see [5]). In
[4] it was shown that $Gr_{A}$ is a complemented submanifold of ${\cal
B}_{A}({\cal H})$, and an homogeneous space of the group ${\cal U}_{A}$ under
the action

$G\cdot{\cal S}=G({\cal S})$

or equivalently

$G\cdot Q_{\cal S}=GQ_{\cal S}G^{-1},\ \ G\in{\cal U}_{A},\ {\cal S}\in
Gr_{A}.$

An $A$-isometry may not be an adjointable operator.

######  Examples 2.4.

  1. 1.

Let ${\cal L}=H^{2}(\mathbb{T})$ the Hardy space of the unit circle
$\mathbb{T}$, ${\cal H}=\displaystyle$ the Dirichlet space of the unit disk
$\mathbb{D}$, i.e.,

${\cal L}=\\{f:\mathbb{D}\to\mathbb{C}:f\hbox{ is analytic and
}\sum_{n=0}^{\infty}|\hat{f}(n)|^{2}<\infty\\},$ ${\cal H}=\\{f\in{\cal
L}:\sum_{n=0}^{\infty}(n+1)|\hat{f}(n)|^{2}<\infty\\}.$

Here the inner product of ${\cal H}$ is

$\langle f,g\rangle=\sum_{n=0}^{\infty}(n+1)\hat{f}(n)\overline{\hat{g}(n)},$

and the inner product of ${\cal L}$ is

$\langle f,g\rangle=\sum_{n=0}^{\infty}\hat{f}(n)\overline{\hat{g}(n)},$

in both cases for $f,g\in{\cal H}$. So that $A\in{\cal B}({\cal H})$ is given
by

$Af(z)=\sum_{n=0}^{\infty}\frac{1}{n+1}\hat{f}(n)z^{n}.$

Note that $A$ is compact. Let ${\bf T}=M_{z}\in{\cal B}({\cal L})$ be the
shift operator. Clearly ${\bf T}({\cal H})\subset{\cal H}$ and ${\bf
T}^{*}({\cal H})\subset{\cal H}$, so that $T={\bf T}|_{\cal H}$ is an
adjointable $A$-isometry. In particular, see below (Theorem 2.6), this means
that $R(T)=z{\cal H}$ is a compatible subspace.

  2. 2.

Consider the slight modification of the above example. Put ${\cal L}=\ell^{2}$
and ${\cal
H}=\\{(x_{n})\in\ell^{2}:\sum_{n=1}^{\infty}n|x_{n}|^{2}<\infty\\}$, with its
natural inner product $\langle(x_{n}),(y_{n})\rangle_{{}_{\cal
H}}=\sum_{n=1}^{\infty}nx_{n}\bar{y}_{n}$. Let $e_{n}$, $n\geq 1$ be the
canonical orthonormal basis of $\ell^{2}$. Let $\mathbb{A}$ the subset of the
natural numbers which are squares of odd integers,

$\mathbb{A}:=\\{(2k+1)^{2}:k\geq 0\\},$

and denote its complement

$\mathbb{A}^{c}=\\{\sigma(1)<\sigma(2)<\sigma(3)<\dots\\}.$

Let $U^{*}$ be the unitary operator in $\ell^{2}$ given by

$U^{*}e_{n}=\left\\{\begin{array}[]{cc}e_{n^{2}}&\hbox{ if }n\hbox{ is
odd,}\\\ e_{\sigma(n/2)}&\hbox{ if }n\hbox{ is even.}\end{array}\right.$

Clearly $\\{U^{*}e_{n}:n\geq 1\\}=\\{e_{n}:n\geq 1\\}$, so that $U^{*}$ is
indeed a unitary operator in $\ell^{2}$. Its adjoint $U$ is given by (for
$x=(x_{n})\in\ell^{2}$)

$(Ux)_{n}=\left\\{\begin{array}[]{cc}x_{n^{2}}&\hbox{ if }n\hbox{ is odd},\\\
x_{\sigma(n/2)}&\hbox{ if }n\hbox{ is even.}\end{array}\right.$

We claim that $U({\cal H})\subset{\cal H}$, but $U^{*}({\cal H})$ is not
contained in ${\cal H}$. Which means that $U|_{\cal H}\in{\cal I}_{A}$, but it
is not $A$-adjointable. For the first assertion, note the fact that for all
$k\geq 1$, $k\leq\sigma(k)\leq 2k$. Then, if $(x_{n})\in{\cal H}$,

$\begin{array}[]{cclcl}\langle Ux,Ux\rangle_{{}_{\cal H}}&=&\sum\limits_{n\
\textnormal{\scriptsize{odd}}}n|x_{n^{2}}|^{2}&+&\sum\limits_{n\
\textnormal{\scriptsize{even}}}n|x_{\sigma(n/2)}|^{2}\\\ &\leq&\sum\limits_{n\
\textnormal{\scriptsize{odd}}}n^{2}|x_{n^{2}}|^{2}&+&\sum\limits_{n\
\textnormal{\scriptsize{even}}}2\sigma(n/2)|x_{\sigma(n/2)}|^{2}\\\
&\leq&2\langle x,x\rangle_{{}_{\cal H}}.&\end{array}$

To prove that ${\cal H}$ is not invariant for $U^{*}$, pick
$x_{k}=\frac{1}{k^{3/2}}$ if $k$ is odd and $x_{k}=0$ otherwise. Then clearly,
$x=(x_{n})\in{\mathcal{H}}$, and

$(U^{*}x)_{n}=\left\\{\begin{array}[]{cc}\frac{1}{n^{3/4}}&\hbox{ if
}n\in\mathbb{A},\\\ 0&\hbox{ if }n\in\mathbb{A}^{c}\end{array}\right.$

Or equivalently, $(U^{*}x)_{(2k+1)^{2}}=\frac{1}{(2k+1)^{3/2}}$ and equal to
$0$ in all other entries. Then

$\langle
U^{*}x,U^{*}x\rangle_{{\mathcal{H}}}=\sum_{k=0}(2k+1)^{2}\left(\frac{1}{(2k+1)^{3/2}}\right)^{2}=\sum_{k=0}\frac{1}{(2k+1)}=+\infty.$

Consider the subclass of $A$-adjointable $A$-isometries, or shortly,
adjointable isometries. Operators $G\in{\cal U}_{A}$ are examples of
adjointable isometries. It is proved below that adjointable isometries are
those with compatible final spaces. It will be useful to recall the following
result by R. Douglas [9]:

######  Remark 2.5.

Douglas’ theorem considers the existence of solutions of the operator equation
$AX=B$ [9]: let $A,B\in{\cal B}({\cal H})$, then the following conditions are
equivalent:

  1. 1.

there exists $X\in{\cal B}({\cal H})$ such that $AX=B$;

  2. 2.

$R(B)\subset R(A)$;

  3. 3.

there exists $\lambda>0$ such that $BB^{*}\leq\lambda AA^{*}$.

######  Theorem 2.6.

Let $T\in{\cal I}_{A}$. Denote by ${\bf T}$ its (isometric) extension to
${\cal L}$. The following are equivalent:

  1. 1.

$T$ is $A$-adjointable;

  2. 2.

${\bf T}^{*}({\cal H})\subset{\cal H}$;

  3. 3.

$R(T)$ is a compatible subspace.

  4. 4.

$R(T^{*}A)=R(A)$;

  5. 5.

there exists $\lambda>0$ such that

$T^{*}A^{2}T\leq\lambda A^{2}.$

###### Proof.

The equivalence between the first two conditions is clear: if $T$ has an
$A$-adjoint $S$, then ${\bf T}^{*}$ and $S$ coincide on ${\cal H}$. Then ${\bf
T}^{*}({\cal H})\subset{\cal H}$ and $S={\bf T}^{*}|_{\cal H}$. If ${\bf
T}^{*}({\cal H})\subset{\cal H}$, then ${\bf T}^{*}|_{\cal H}$ is the
$A$-adjoint of $T$. Note that therefore the final projection ${\bf P}={\bf
T}{\bf T}^{*}$ of ${\bf T}$ leaves ${\cal H}$ invariant: ${\bf P}({\cal
H})\subset{\cal H}$. Thus, ${\bf P}$ induces an ($A$-symmetric) idempotent
$P={\bf P}|_{\cal H}$ in ${\cal H}$. Clearly, $P=TS$, and $R(P)=R(T)$: $P=TS$
implies that $R(P)\subset R(T)$; on the other hand, ${\bf T}{\bf T}^{*}{\bf
T}={\bf T}$, so that, restricting to ${\cal H}$, $TST=PT=T$ ans therefore
$R(T)\subset R(P)$. Summarizing, $R(T)$ is the range of an $A$-symmetric
projection, i.e., a compatible subspace.

Conversely, suppose that $R(T)$ is compatible and denote by $P$ the unique
$A$-symmetric idempotent with $R(P)=R(T)$. In particular, $R(T)$ is closed,
and $T$ has a pseudo-inverse $S$ such that $TS=P$. Indeed,
$T|_{N(T)^{\perp}}\to R(T)$ is an isomorphism between Banach spaces, thus it
has a bounded inverse $T^{\prime}$. Put $S$

$S=\left\\{\begin{array}[]{l}T^{\prime}\hbox{ in }R(P)=R(T)\\\ 0\hbox{ in
}N(P)\end{array}\right..$

Then $TS$ equals the identity in $R(P)$ and is zero in $N(P)$, i.e., $TS=P$.
Note that $PT=T$, and that $P^{*}A=AP$. Then $T^{*}AT=A$ implies that

$AS=T^{*}ATS=T^{*}AP=T^{*}P^{*}A=(PT)^{*}A=T^{*}A,$

i.e., $S$ is the $A$-adjoint of $T$.

The equivalence with the last two conditions follows using Douglas’ result:
$T$ is $A$-adjointable if and only if there exists a solution $X$ to the
operator equation $AX=T^{*}A$ (i.e., $B=T^{*}A$), which occurs if and only if
$R(T^{*}A)\subset R(A)$, or equivalently, there exists $\lambda>0$ such that

$BB^{*}=T^{*}A^{2}T\leq\lambda A^{2}.$

The former condition $R(T^{*}A)\subset R(A)$ is in turn equivalent to

$R(A)=R(T^{*}AT)\subset R(T^{*}A)\subset R(A),$

i.e., $R(A)=R(T^{*}A)$. ∎

######  Remark 2.7.

Note that if $T\in{\cal I}_{A}^{a}$, then $T$ is injective and has closed
range, i.e. $T$ is bounded from below. Note the following example, of an
operator in ${\cal I}_{A}$, with closed range, but whose range is not a
compatible subspace of ${\cal H}$.

######  Example 2.8.

Let ${\cal H}=H_{0}^{1}(0,1)$, the subspace of the Sobolev space $H^{1}(0,1)$
obtained as the closure of the smooth functions in $(0,1)$ with compact
support. Let ${\cal L}=L^{2}(0,1)$. With the same argument as in [4] (Example
3.7), it can be shown that

${\cal S}=\\{f\in{\cal H}_{0}^{1}(0,1):f\equiv 0\hbox{ in }[1/2,1)\\}$

is a (closed) non compatible subspace of ${\cal H}\subset{\cal L}$. Let us
sketch how this is proved. Let $f_{0}$ be a $C^{\infty}$ function in $(0,1)$,
of compact support, which equals $1$ on an interval centered a
$t=\frac{1}{2}$. Then $f_{0}=f_{0}\ \chi_{(0,1/2)}+f_{0}\ \chi_{[1/2,1)}$ is
an orthogonal sum in ${\cal L}$, with $f_{0}\ \chi_{(0,1/2)}\in\overline{{\cal
S}}$ (the closure of ${\cal S}$ in ${\cal L}$). Thus ${\bf P}_{\overline{{\cal
S}}}(f_{0})=f_{0}\ \chi_{(0,1/2)}$, which does no belong to ${\cal H}$, i.e.,
${\bf P}_{\overline{{\cal S}}}({\cal H})\not\subset{\cal H}$, and ${\cal S}$
is not compatible. Consider the operator ${\bf V}$ in ${\cal B}({\cal L})$
given by

${\bf V}f(t)=\left\\{\begin{array}[]{ll}\sqrt{2}\ f(2t)&\hbox{ if
}t\in(0,1/2)\\\ 0&\hbox{ if }t\in[1/2,1).\end{array}\right.$

Note that

$\|{\bf V}f\|_{2}=\int_{0}^{1/2}|{\bf V}f(t)|^{2}dt\stackrel{{\scriptstyle
2t=s}}{{=}}\int_{0}(\sqrt{2})^{2}|f(s)|^{2}\frac{1}{2}ds=\|f\|^{2},$

i.e. ${\bf V}$ is an isometry of ${\cal L}$. Clearly ${\bf V}$ preserves
smooth funcions of compact support, ${\bf V}({\cal H})\subset{\cal H}$. Also
it is clear that ${\bf V}({\cal H})$ consists of all functions in $H^{1}(0,1)$
with compact support contained in $(0,1/2)$, i.e., ${\bf V}({\cal H})={\cal
S}$. Thus, $V={\bf V}|_{\cal H}$ belongs to ${\cal I}_{A}$, has closed range,
but is not adjointable.

######  Examples 2.9.

  1. 1.

Example 2.4.1, where $T$ is the shift operator and ${\cal H}\subset{\cal L}$
are, respectively, the Dirichlet and the Hardy space of the circle, can be
generalized. Let $T\in{\cal I}_{A}$ such that $R(T)$ is closed and has finite
co-dimension. Then $T\in{\cal I}_{A}^{a}$. Indeed, in [4] it was shown that
closed subspaces with finite co-dimension are compatible.

  2. 2.

There are also examples of $T\in{\cal I}_{A}^{a}$ where $R(T)$ has infinite
co-dimension. For instance, consider the shift (of infinite multiplicity)
${\bf T}$ in ${\cal L}=\ell^{2}$, ${\bf T}e_{n}=e_{2n}$. Consider as before,
${\cal H}=\\{(x_{n})\in\ell^{2}:\sum_{n=1}^{\infty}n|x_{n}|^{2}<\infty\\}$.
Clearly ${\bf T}({\cal H})\subset{\cal H}$. Note that $({\bf
T}^{*}x)_{n}=x_{2n}$, and then

$\langle{\bf T}^{*}x,{\bf T}^{*}x\rangle_{\cal
H}=\sum_{n=1}^{\infty}n|x_{2n}|^{2}<\sum_{n=1}^{\infty}2n|x_{2n}|^{2}\leq\langle
x,x\rangle_{\cal H},$

i.e., ${\bf T}^{*}({\cal H})\subset{\cal H}$. Then $T={\bf T}|_{\cal
H}\in{\cal I}_{A}^{a}$.

######  Remark 2.10.

Again, using Douglas’ result, it holds that $T\in{\cal I}_{A}$ is always
$A^{1/2}$-adjointable:

$T^{*}(A^{1/2})^{2}T=T^{*}AT=A=(A^{1/2})^{2},$

which is the second condition of Douglas for $\lambda=1$.

Note the evident facts that if $T\in{\cal I}_{A}^{a}$, then $T^{n}\in{\cal
I}_{A}^{a}$ for $n\geq 1$; also, $T^{\sharp}T=1$ in ${\cal H}$.

##  3 Wold decomposition of $A$-adjointable isometries

Recall the Wold decomposition of an isometry (see for instance [14]): given an
isometry ${\bf V}$ acting in ${\cal L}$, there exists an orthogonal
decomposition

${\cal L}={\cal L}_{0}\oplus{\cal L}_{1}$

such that ${\cal L}_{0},{\cal L}_{1}$ reduce ${\bf V}$, ${\bf V}|_{{\cal
L}_{0}}$ is a unitary operator and ${\bf V}|_{{\cal L}_{1}}$ is a shift
(${\cal L}_{1}=\bigoplus\limits_{n=0}^{\infty}{\bf V}^{n}{\cal L}_{w}$, where
${\cal L}_{w}={\cal L}\ominus{\bf V}{\cal L}$ is the so called wandering space
of ${\bf V}$). This decomposition is unique, ${\cal L}_{1}$ is determined by
${\bf V}$, as seen in the above formula; also ${\cal
L}_{0}=\bigcap_{n=0}^{\infty}{\bf V}^{n}{\cal L}$.

The next result shows that the Wold decomposition of ${\bf V}$ in ${\cal L}$
induces an analogous compatible decomposition for $V$ in ${\cal H}$.

######  Theorem 3.1.

Let $V\in{\cal I}_{A}^{a}$ and ${\bf V}$ its extension to an isometry of
${\cal L}$, then the Wold decomposition ${\cal L}={\cal L}_{0}\oplus{\cal
L}_{1}$ of ${\bf V}$ induces a (direct sum, $A$-orthogonal) decomposition

${\cal H}={\cal H}_{0}\oplus{\cal H}_{1},\ {\cal H}_{0}={\cal L}_{0}\cap{\cal
H},\ {\cal H}_{1}={\cal H}\cap{\cal L}_{1},$

of compatible subspaces. $V{\cal H}_{0}={\cal H}_{0}$, $V{\cal
H}_{1}\subset{\cal H}_{1}$, so that $V|_{{\cal H}_{0}}$ is invertible in
${\cal H}_{0}$.

###### Proof.

The wandering subspace ${\cal L}_{w}={\cal L}\ominus{\bf V}{\cal L}$ is the
orthogonal complement of the range of ${\bf V}$, therefore its intersection
with ${\cal H}$ is the $A$-orthogonal complement $R(V)^{\perp_{A}}$ of $R(V)$
(which are both compatible subspaces of ${\cal H}$). Note that for $n\geq 1$,

$V^{n}(R(V)^{\perp_{A}})={\bf V}^{n}(R(V)^{\perp_{A}})\hbox{ is dense in }{\bf
V}^{n}{\cal L}_{w}.$

Thus the subspaces $V^{n}(R(V)^{\perp_{A}})$ are in direct sum for different
$n$, and their sum

$\bigoplus_{n=0}^{\infty}V^{n}(R(V)^{\perp_{A}})\hbox{ is dense in
}\bigoplus_{n=0}^{\infty}{\bf V}^{n}{\cal L}_{w}.$

Note also that since $V^{n}\in{\cal I}_{A}^{a}$, $V^{n}(R(V)^{\perp_{A}})$ is
a compatible subspace (in particular, closed). Therefore the $A$-orthogonal
sum (and direct sum in ${\cal H}$)
$\bigoplus_{n=0}^{\infty}V^{n}(R(V)^{\perp_{A}})$ is a compatible subspace.

Next, ${\bf V}{\cal L}_{0}={\cal L}_{0}$, and thus $V({\cal H}_{0})\subset{\bf
V}{\cal L}_{0}={\cal L}_{0}$, together with $V({\cal H}_{0})\subset{\cal H}$,
implies $V({\cal H}_{0})\subset{\cal H}\cap{\cal L}_{0}={\cal H}_{0}$. By the
same reason, since $V^{\sharp}={\bf V}^{*}|_{\cal H}$. Also $V^{\sharp}({\cal
H}_{0})\subset{\cal H}_{0}$. Moreover,

${\cal H}_{0}=V^{\sharp}V({\cal H}_{0})\subset V^{\sharp}({\cal
H}_{0})\subset{\cal H}_{0}.$

Since clearly ${\cal H}_{0}\subset R(V)$, then $VV^{\sharp}=P_{V}$ (the
$A$-symmetric idempotent in ${\cal H}$ with range equal to $R(V)$) satisfies
$VV^{\sharp}{\cal H}_{0}={\cal H}_{0}$. Then

${\cal H}_{0}=VV^{\sharp}({\cal H}_{0})\subset V({\cal H}_{0})\subset{\cal
H}_{0}.$

That is, $V({\cal H}_{0})={\cal H}_{0}$, i.e. $V|_{{\cal H}_{0}}$ is
invertible. ∎

######  Remark 3.2.

In the above theorem it was shown that $V|_{{\cal H}_{0}}$ is invertible in
${\cal H}_{0}$. The operator $A$ does not necessarily leave ${\cal H}_{0}$
invariant, nevertheless $A$ induces an inner product in ${\cal H}_{0}$. For
this induced inner product (which is implemented by the compression of $A$ to
${\cal H}_{0}$: if $f,g\in{\cal H}_{0}$, $\langle f,g\rangle_{A}=\langle
Af,g\rangle=\langle AP_{{\cal H}_{0}}f,P_{{\cal H}_{0}}g\rangle=\langle
P_{{\cal H}_{0}}AP_{{\cal H}_{0}}f,g\rangle$), the restriction $V|_{{\cal
H}_{0}}$ is a isometric and onto, i.e. $V|_{{\cal H}_{0}}$ belongs to the
group ${\cal U}_{P_{{\cal H}_{0}}AP_{{\cal H}_{0}}}$ of the Hilbert space
${\cal H}_{0}$.

##  4 Regular structure of ${\cal I}_{A}^{a}$

In the introduction it was observed that ${\cal U}_{A}$ acts on ${\cal
I}_{A}^{a}$: if $G\in{\cal U}_{A}$ and $T\in{\cal I}_{A}^{a}$, then
$GT\in{\cal I}_{A}^{a}$, being a composition of $A$-adjointable isometries.

Let us recall in the next remark, certain facts on $A$-orthogonal projections
(see [4])

######  Remark 4.1.

Let

${\cal P}_{A}=\\{P\in{\cal B}({\cal H}):P^{2}=P,P\hbox{ is
}A-\hbox{symmetrizable}\\}.$

Recall that ${\cal B}_{A}({\cal H})$ denotes the algebra of operators acting
in ${\cal H}$ that are $A$-adjointable. It is an involutive Banach algebra
with the norm $|T|=\max\\{\|T\|,\|T^{\sharp}\|\\}$. Clearly
$|T^{\sharp}|=|T|$. The set ${\cal P}_{A}$ is a complemented
$C^{\infty}$-submanifold of ${\cal B}_{A}({\cal H})$, and a homogeneous space
of ${\cal U}_{A}$ (which in turn is a Banach-Lie group and a $C^{\infty}$
submanifold of ${\cal B}_{A}({\cal H})$). This means that for each
$P_{0}\in{\cal P}_{A}$, the map $\pi_{P_{0}}:{\cal U}_{A}\to{\cal P}_{A}$,
$\pi_{P_{0}}(G)=GP_{0}G^{-1}$, is a submersion: it is surjective onto the
orbit of $P_{0}$ by the action, which is a union of connected components. In
particular, it has smooth local cross sections. That is, there exists a radius
$r_{P_{0}}>0$ and a map $\sigma_{P_{0}}$,

$\sigma_{P_{0}}:\\{P\in{\cal O}_{A}:|P-P_{0}|<r_{P_{0}}\\}\subset{\cal
P}_{A}\to{\cal U}_{A},$

with the following properties:

  * •

$\sigma_{P_{0}}$ is a $C^{\infty}$ (local) cross section of $\pi_{P_{0}}$:

$\pi_{P_{0}}(\sigma_{P_{0}}(P))=\sigma_{P_{0}}(P)P_{0}\sigma_{P_{0}}(P)^{-1}=P,\hbox{
for }P\in{\cal P}_{A}\hbox{ with }|P-P_{0}|<r_{P_{0}};$

  * •

the map $\sigma_{P_{0}}$ extends to an open ball of ${\cal B}_{a}({\cal H})$
centered at $P_{0}$, as a $C^{\infty}$ map with values in the invertible group
of ${\cal B}_{a}({\cal H})$;

  * •

if one moves $P$ and $P_{0}$, locally, the element $\sigma_{P_{0}}(P)$ is
$C^{\infty}$ in both variables.

The first two conditions, in fact, imply that $\pi_{P_{0}}$ is a submersion
and ${\cal P}_{A}$ a complemented submanifold:

######  Lemma 4.2.

Let ${\cal A}$ be a Banach algebra, ${\cal U}$ be a Banach-Lie subgroup of the
invertible group ${\cal A}^{\times}$ of ${\cal A}$, and $X$ a Banach space on
which ${\cal U}$ acts smoothly. Let $x_{0}\in X$ be a fixed element. Suppose
that the map

$\pi_{x_{0}}:{\cal U}\to{\cal O}_{x_{0}}:=\\{u\cdot x_{0}:u\in{\cal U}\\}$

has a continuous cross section $\sigma_{x_{0}}$ defined on an neighbourhood of
$x_{0}$ in ${\cal O}_{x_{0}}$ in the relative topology induced by $X$. If
$\sigma_{x_{0}}$ can be extended to a smooth map in an open neighbourhood of
$x_{0}$ in $X$, then ${\cal O}_{x_{0}}$ is a complemented smooth submanifold
of $X$ and the map $\pi_{x_{0}}$ is a submersion.

###### Proof.

The hypothesis imply that $\pi_{x_{0}}$ is open as a map ${\cal U}\to{\cal
O}_{x_{0}}$, and is a smooth map that splits, regarded as a map ${\cal U}\to
X$. ∎

Let us fix $T_{0}\in{\cal I}_{A}^{a}$, and construct the extendable local
cross sections of

$\pi_{T_{0}}:{\cal U}_{A}\to{\cal I}_{A}^{a},\ \pi_{T_{0}}(G)=GT_{0}.$ (3)

For $T\in{\cal I}_{A}^{a}$, denote by $P_{T}:=TT^{\sharp}$. Clearly $P_{T}$ is
an $A$-symmetric projection, which extends to the orthogonal projection ${\bf
P}_{R({\bf T})}$ onto the range of ${\bf T}$ in ${\cal L}$.

If $T$ is close to $T_{0}$, then $P_{T}$ is close in $P_{T_{0}}$. Explicitly,

$|P_{T}-P_{T_{0}}|=|TT^{\sharp}-T_{0}T_{0}^{\sharp}|\leq|TT^{\sharp}-TT_{0}^{\sharp}|+|TT_{0}^{\sharp}-T_{0}T_{0}^{\sharp}|\leq|T||T^{\sharp}-T_{0}^{\sharp}|+|T-T_{0}||T_{0}^{\sharp}|$
$=|T-T_{0}|(|T|+|T_{0}|)\leq|T-T_{0}|(|T-T_{0}|+2|T_{0}|),$

In particular, there exists $\delta_{T_{0}}$, which depends on $T_{0}$, such
that if $|T-T_{0}|<\delta_{T_{0}}$, then $|P_{T}-P_{T_{0}}|<r_{P_{T_{0}}}$
(the radius given in the above remark).

Suppose that $|T-T_{0}|<\delta_{T_{0}}$, and denote by
$G_{T}=\sigma_{P_{T_{0}}}(P_{T})\in{\cal U}_{A}$. Clearly $G_{T}$ is a
$C^{\infty}$ map of $T$ which satisfies that
$G_{T}P_{T_{0}}G_{T_{0}}^{-1}=P_{T}$. Then $T^{\prime}:=G_{T}^{-1}T$ is an
element of ${\cal I}_{A}^{a}$ which has final projection

$T^{\prime}T^{\prime\sharp}=G_{T}^{-1}TT^{\sharp}(G^{-1})^{\sharp}=G_{T}^{-1}P_{T}G_{T}=P_{T_{0}}.$

Two elements $T^{\prime},T_{0}$ of ${\cal I}_{A}^{a}$ with the same final
projection $P_{T_{0}}$ are conjugate by the action of ${\cal U}_{A}$: there
exists $H=H_{T_{0}}(T^{\prime})\in{\cal U}_{A}$ such that $HT_{0}=T^{\prime}$.
Pick, for instance

$H=T^{\prime}T_{0}^{\sharp}+(1-P_{T_{0}}).$

Note that

$\begin{array}[]{ccl}HH^{\sharp}&=&T^{\prime}T_{0}^{\sharp}T_{0}T^{\prime\sharp}+T^{\prime}T_{0}^{\sharp}(1-P_{T_{0}})+(1-P_{T_{0}})T_{0}T^{\prime\sharp}+(1-P_{T_{0}})\\\
&=&T^{\prime}T^{\prime\sharp}+(1-P_{T_{0}})=P_{T_{0}}+1-P_{T_{0}}\\\
&=&1,\end{array}$

because $(1-P_{T_{0}})T^{\prime}=T_{0}^{\sharp}(1-P_{T_{0}})=0$. Similarly
$H^{\sharp}H=1$. Moreover,

$HT_{0}=T^{\prime}T_{0}^{\sharp}T_{0}+(1-P_{T_{0}})T_{0}=T^{\prime}.$

Clearly $H=H_{T_{0}}(T^{\prime})$ is a $C^{\infty}$ map in terms of
$T^{\prime}$ and $T_{0}$, and thus a $C^{\infty}$ map in terms of $T$.

Thus, $H\in{\cal U}_{A}$ and $HT_{0}=G_{T}^{-1}T$, i.e., $T=G_{T}HT_{0}$.
Define

$\sigma_{T_{0}}(T)=G_{T}H=G_{T}(G_{T}^{-1}TT_{0}^{\sharp}+(1-P_{T_{0}})=TT_{0}^{\sharp}+G_{T}(1-P_{T_{0}}),\
\hbox{ for }T\in{\cal I}_{A}^{a},\ |T-T_{0}|<\delta_{T_{0}}.$ (4)

######  Theorem 4.3.

Let $T_{0}\in{\cal I}_{A}^{a}$. Then the orbit

${\cal O}_{T_{0}}:=\\{GT_{0}:G\in{\cal U}_{A}\\}$

is a $C^{\infty}$ complemented submanifold of ${\cal B}_{A}({\cal H})$, and
the map

$\pi_{T_{0}}:{\cal U}_{A}\to{\cal O}_{T_{0}},\ \pi_{T_{0}}(G)=GT_{0}$

is a $C^{\infty}$ submersion. The orbit ${\cal O}_{T_{0}}$ is a union of
connected components of ${\cal I}_{A}^{a}$, which implies that ${\cal
I}_{A}^{a}$ is a $C^{\infty}$ complemented submanifold of ${\cal B}_{A}({\cal
H})$

###### Proof.

The fact that ${\cal O}_{T_{0}}$ is a submanifold follows using Lemma 4.2,
noting that the cross section $\sigma_{T_{0}}$ defined in (4) clearly extends
to a $C^{\infty}$ map defined on a neighbourhood of $T_{0}$ in ${\cal
B}_{A}({\cal H})$. Let us check fact that ${\cal O}_{T_{0}}$ is a union of
connected components of ${\cal I}_{A}^{a}$. In [4] it was observed that the
local cross section $P_{T}\mapsto G_{T}$ for $\pi_{P_{0}}$ satisfies that that
$G_{T_{0}}=1$, and therefore $G_{T}$ belongs to the connected component of the
identity in ${\cal U}_{A}$ if $T$ is close to $T_{0}$. This implies that if
$T$ is close enough to $T_{0}$, then $\sigma_{T_{0}}(T)$ belongs to the
connected component of the identity of ${\cal U}_{A}$. This observation
implies that any $T$ close to $T_{0}$, can be connected to $T_{0}$ by means of
a continuous path in the orbit. ∎

##  5 The extension method of M.G. Krein for
symmetrizable operators

In this section the extension method of M.G. Krein [11] is considered (see
also Section 125 of the classic book [16] for a detailed exposition). Krein
shows that if $S:{\cal L}_{0}\subset{\cal L}\to{\cal L}$ is a contractive
symmetric operator, then there exists a symmetric extension $S:{\cal
L}\to{\cal L}$ which is also a contraction. Later on, other authors elaborated
on this problem (see for instance [8] were the solutions are parametrized, or
the works of Parrott [15]). In this section, following the ideas of M.G. Krein
with slight modifications, it will be shown that an $A$-symmetric operator
defined on a $A$-compatible subspace ${\cal H}_{0}\subset{\cal H}$, which is
contractive for the norm induced by $\langle\ \ ,\ \ \rangle_{A}$, can be
extended to a contraction for this norm to the whole space ${\cal H}$. Or
rather, and equivalently, we shall work in the bigger Hilbert space ${\cal
L}$, extending to a symmetric contraction of ${\cal L}$ which leaves ${\cal
H}$ invariant.

######  Lemma 5.1.

Let ${\bf X}$ be a selfadjoint operator in ${\cal L}$ such that ${\bf X}({\cal
H})\subset{\cal H}$, and let ${\bf P}$ be a projection in ${\cal L}$ with
range ${\cal L}_{0}$, such that $P={\bf P}|_{\cal H}$ is an idempotent in
${\cal H}$ with range ${\cal H}_{0}$. Suppose that $\|{\bf X}{\bf P}\|=1$.
Then there exists a selfadjoint operator ${\bf Z}$ in ${\cal L}$ such that
${\bf Z}({\cal H})\subset{\cal H}$, ${\bf Z}{\bf P}={\bf X}{\bf P}$ and
$\|{\bf Z}\|=1$.

###### Proof.

For $m>0$, let us denote by ${\bf X}_{m}=\frac{1}{m}{\bf X}$, $m$ will be
adjusted in the process. Denote by ${\cal H}_{0}={\bf P}({\cal H})$. Let ${\bf
B}_{0}={\bf P}{\bf X}_{m}{\bf P}:{\cal L}_{0}\to{\cal L}_{0}$. Then ${\bf
B}_{0}$ extends to an operator in ${\cal L}$ with the same norm: $\bar{{\bf
B}}_{0}={\bf P}{\bf X}_{m}:{\cal L}\to{\cal L}_{0}$. Note that $\bar{{\bf
B}}_{0}$ leaves ${\cal H}$ invariant, because ${\bf X}_{m}$ and ${\bf P}$ do.

Next extend ${\bf B}_{1}:={\bf P}^{\perp}{\bf X}_{m}{\bf P}:{\cal
L}_{0}\to{\cal L}_{0}^{\perp}$ to the whole space ${\cal L}$ (not enlarging
the norm of ${\bf B}_{1}$ and leaving ${\cal H}$ invariant). Consider the
inner product

$[\varphi,\gamma]:=\langle\varphi,\gamma\rangle_{\cal L}-\langle\bar{{\bf
B}}_{0}\varphi,\bar{{\bf B}}_{0}\gamma\rangle_{\cal L}.$

Note that

$[\varphi,\gamma]:=\langle\varphi,\gamma\rangle_{\cal L}-\langle{\bf P}{\bf
X}_{m}\varphi,{\bf P}{\bf X}\gamma\rangle_{\cal L}=\langle(I-{\bf X}_{m}{\bf
P}{\bf X}_{m})\varphi,\gamma\rangle_{\cal L}.$

Take $m>0$ so that $\|{\bf X}_{m}{\bf P}{\bf X}_{m}\|<1$, then $I-{\bf
X}_{m}{\bf P}{\bf X}_{m}$ is (positive and) invertible. Then $({\cal L},[\ \
,\ \ ])$ is a Hilbert space, whose norm is equivalent to the original norm
induced by $\langle\ \ ,\ \ \rangle_{\cal L}$. The operator ${\bf B}_{1}$ is
bounded by (for $\varphi={\bf P}\varphi\in{\cal L}_{0}$)

$\|{\bf B}_{1}\varphi\|^{2}=\langle{\bf P}^{\perp}{\bf X}_{m}{\bf
P}\varphi,{\bf P}^{\perp}{\bf X}_{m}{\bf P}\varphi\rangle_{\cal L}=\langle{\bf
X}_{m}(1-{\bf P}){\bf X}_{m}{\bf P}\varphi,{\bf P}\varphi\rangle_{\cal L}$
$=\langle{\bf X}_{m}^{2}{\bf P}\varphi,{\bf P}\varphi\rangle_{\cal
L}-\langle{\bf X}_{m}{\bf P}{\bf X}_{m}{\bf P}\varphi,{\bf
P}\varphi\rangle_{\cal L}.$

Since $\|{\bf X}{\bf P}\|=\frac{1}{m}$, the first term is bounded by

$\langle{\bf X}_{m}^{2}{\bf P}\varphi,{\bf P}\varphi\rangle_{\cal
L}=\langle{\bf P}{\bf X}_{m}^{2}{\bf P}\varphi,\varphi\rangle_{\cal
L}\leq\|{\bf P}{\bf X}_{m}^{2}{\bf
P}\|\|\varphi\|^{2}=\frac{1}{m^{2}}\|\varphi\|^{2}.$

Pick $m$ so that $m\geq 1$, one has that

$\|{\bf
B}_{1}\varphi\|^{2}\leq\frac{1}{m^{2}}\langle\varphi,\varphi\rangle_{\cal
L}-\langle{\bf X}_{m}{\bf P}{\bf X}_{m}\varphi,\varphi\rangle_{\cal
L}\leq\frac{1}{m^{2}}\langle\varphi,\varphi\rangle_{\cal
L}-\frac{1}{m^{2}}\langle{\bf X}_{m}{\bf P}{\bf
X}_{m}\varphi,\varphi\rangle_{\cal L}=\frac{1}{m^{2}}[\varphi,\varphi].$

Therefore ${\bf B}_{1}$ induces an operator ${\cal B}:({\cal L}_{0},[\ \ ,\ \
])\to({\cal L}_{0}^{\perp},\langle\ \ ,\ \ \rangle_{\cal L})$ with norm less
than or equal to $\frac{1}{m}$. Since at the set level, this mapping coincides
with ${\bf B}_{1}$, ${\cal B}$ maps ${\cal H}_{0}$ into ${\cal
L}_{0}^{\perp}\cap{\cal H}$. Let $\Pi:({\cal L},[\ \ ,\ \ ])\to({\cal
L}_{0},[\ \ ,\ \ ])$ denote the $[\ \ ,\ \ ]$-orthogonal projection. Since $[\
\ ,\ \ ]=\langle(1-{\bf X}_{m}{\bf P}{\bf X}_{m})\ \ ,\ \ \rangle_{\cal L}$,
the $[\ \ ,\ \ ]$ adjoint of an operator ${\cal T}$ acting in $({\cal L},[\ \
,\ \ ])$ is given by ${\cal T}^{\sharp}=(1-{\bf X}_{m}{\cal T}^{*}{\bf
X}_{m})^{-1}{\cal T}^{*}(1-{\bf X}_{m}{\cal T}^{*}{\bf X}_{m})$, with ${\cal
T}^{*}$ the $\langle\ \ ,\ \ \rangle_{\cal L}$\- adjoint of ${\cal T}$. It is
known (see for instance [1]), that the orthogonal projection onto the range of
an idempotent $Q$ in a Hilbert space, is given by the formula

$P_{R(Q)}=Q(Q+Q^{*}-1)^{-1}.$

In our case, this implies that

$\Pi={\bf P}\\{{\bf P}+(1-{\bf X}_{m}{\bf P}{\bf X}_{m})^{-1}{\bf P}(1-{\bf
X}_{m}{\bf P}{\bf X}_{m})-1\\}^{-1}.$

If one further adjusts $m>0$, one has that $\Pi({\cal H})\subset{\cal H}$. To
shorten the writing, denote by $Q_{m}$ the idempotent $(1-{\bf X}_{m}{\bf
P}{\bf X}_{m})^{-1}{\bf P}(1-{\bf X}_{m}{\bf P}{\bf X}_{m})$. First one needs
that $Q_{m}({\cal H})\subset{\cal H}$. Since $1-{\bf X}_{m}{\bf P}{\bf X}_{m}$
leaves ${\cal H}$ invariant, it induces an operator in ${\cal H}$. If one
further enlarges $m$ so that $X_{m}PX_{m}={\bf X}_{m}{\bf P}{\bf X}_{m}|_{\cal
H}$ has norm strictly less than $1$ in ${\cal B}({\cal H})$, $1-X_{m}PX_{m}$
will be invertible in ${\cal B}({\cal H})$, i.e. $(1-{\bf X}_{m}{\bf P}{\bf
X}_{m})^{-1}$ leaves ${\cal H}$ invariant. It follows that $Q_{m}({\cal
H})\subset{\cal H}$.

Next, note the elementary identity

$({\bf P}+Q_{m}-1)^{2}=1-({\bf P}-Q_{m})^{2},$ (5)

which only uses the fact that ${\bf P}$ and $Q_{m}$ are idempotents. In view
of (5), in order to to have that $({\bf P}+Q_{m}-1)^{2}$, and therefore also
${\bf P}+Q_{m}-1$, is invertible, it suffices to adjust $m$ so that the norm
of ${\bf P}-Q_{m}$ as an operator restricted to ${\cal H}$, is strictly less
than $1$. This clearly can be done, since $X_{m}PX_{m}\to 0$ in ${\cal
B}({\cal H})$ as $m\to+\infty$. It follows that $\Pi({\cal H})\subset{\cal
H}$, for $m$ sufficiently large.

Denote by $J:({\cal L},\langle\ \ ,\ \ \rangle_{\cal L})\to({\cal L},[\ \ ,\ \
])$ the identity mapping. Note that $J$ is contractive:

$[\varphi,\varphi]=\langle(1-{\bf X}_{m}{\bf P}{\bf
X}_{m})\varphi,\varphi\rangle_{\cal L}\leq\langle\varphi,\varphi\rangle_{\cal
L},$

because $\|{\bf X}_{m}\|<1$. Clearly $J({\cal H})\subset{\cal H}$. Therefore,
if one puts

$\bar{{\bf B}}_{1}:={\cal B}\Pi J:{\cal L}\to{\cal L}_{0}^{\perp},$

where both spaces are considered with their original inner product $\langle\ \
,\ \ \rangle_{\cal L}$, one has that (since $\Pi$ and $J$ are contractive),

$\|\bar{{\bf B}}_{1}\|\leq\frac{1}{m}.$

Note also that all the operators involved in this product leave ${\cal H}$
invariant, thus $\bar{{\bf B}_{1}}({\cal H})\subset{\cal H}$. Put

$\bar{{\bf B}}=\bar{{\bf B}}_{0}+\bar{{\bf B}}_{1}.$

It is easily verified that for $\varphi\in{\cal L}$ with $\|\varphi\|=1$,

$\|\bar{{\bf B}}\varphi\|^{2}=\|\bar{{\bf B}}_{0}\varphi\|^{2}+\|\bar{{\bf
B}}_{1}\varphi\|^{2}\leq\frac{1}{m^{2}},$

since $\|\bar{{\bf B}}_{0}\|,\|\bar{{\bf B}}_{1}\|\leq\frac{1}{m}$,
$R(\bar{{\bf B}}_{0})\subset{\cal L}_{0}$ and $R(\bar{{\bf
B}}_{1})\subset{\cal L}_{0}^{\perp}$. Note that $\bar{{\bf B}}_{0}{\bf P}={\bf
P}{\bf X}_{m}{\bf P}$ and that if $\varphi_{0}\in{\cal L}_{0}=R({\bf P})$,

$\bar{{\bf B}}_{1}\varphi_{0}={\cal B}\Pi J\varphi_{0}={\cal
B}\varphi_{0}={\bf B}_{1}\varphi_{0},$

because $\varphi_{0}\in R(\Pi)$. Thus $\bar{{\bf B}}_{1}{\bf P}={\bf
B}_{1}{\bf P}={\bf P}^{\perp}{\bf X}_{m}{\bf P}$. It follows that

$\bar{{\bf B}}{\bf P}=\bar{{\bf B}}_{0}{\bf P}+\bar{{\bf B}}_{1}{\bf P}={\bf
P}{\bf X}_{m}{\bf P}+{\bf P}^{\perp}{\bf X}_{m}{\bf P}={\bf X}_{m}{\bf P}.$

Clearly $\bar{{\bf B}}({\cal H})\subset{\cal H}$. The last thing to fix is
that $\bar{{\bf B}}$ is not selfadjoint.

Note that

${\bf P}\bar{{\bf B}}={\bf P}\bar{{\bf B}}_{0}={\bf P}{\bf X}_{m},$

and then $\bar{{\bf B}}^{*}{\bf P}={\bf X}_{m}{\bf P}$.

${\bf Z}_{m}:=\frac{1}{2}(\bar{{\bf B}}+\bar{{\bf B}}^{*})$

does the feat: ${\bf Z}_{m}^{*}={\bf Z}_{m}$, ${\bf Z}_{m}{\bf P}={\bf
X}_{m}{\bf P}$, $\|{\bf Z}_{m}\|\leq\|\bar{{\bf B}}\|$. In this case one needs
further to verify that $\bar{{\bf B}}^{*}({\cal H})\subset{\cal H}$. Note that

$\bar{{\bf B}}^{*}={\bf X}_{m}{\bf P}+J^{*}\Pi^{*}{\cal B}^{*},$

with the adjoints taken in their respective spaces. Since $\Pi$ is an
orthogonal projection, $\Pi^{*}=\Pi$. Note that the adjoint of $J$ is given by

$\langle\varphi,J^{*}\eta\rangle_{\cal L}=[J\varphi,\eta]=\langle(1-{\bf
X}_{m}{\bf P}{\bf X}_{m})\varphi,\eta\rangle_{\cal L}=\langle\varphi,(1-{\bf
X}_{m}{\bf P}{\bf X}_{m})\eta\rangle_{\cal L},$

i.e., $J^{*}:({\cal L},[\ \ ,\ \ ])\to({\cal L},\langle\ \ ,\ \ \rangle_{\cal
L})$ is $J^{*}=(1-{\bf X}_{m}{\bf P}{\bf X}_{m})$. The adjoint of

${\cal B}:({\cal L}_{0},[\ \ ,\ \ ])\to({\cal L}_{0}^{\perp},\langle\ \ ,\ \
\rangle_{\cal L})$

is given by

$[{\cal B}^{*}\varphi,\eta]=\langle\varphi,{\cal B}\eta\rangle_{\cal
L}=\langle\varphi,{\bf P}^{\perp}{\bf X}_{m}{\bf P}\eta\rangle_{\cal
L}=\langle(1-{\bf X}_{m}{\bf P}{\bf X}_{m})^{-1}(1-{\bf X}_{m}{\bf P}{\bf
X}_{m})\varphi,{\bf P}^{\perp}{\bf X}_{m}{\bf P}\eta\rangle_{\cal L}$
$=[\varphi,(1-{\bf X}_{m}{\bf P}{\bf X}_{m})^{-1}{\bf P}^{\perp}{\bf
X}_{m}{\bf P}\eta],$

i.e., ${\cal B}^{*}=(1-{\bf X}_{m}{\bf P}{\bf X}_{m})^{-1}{\bf P}^{\perp}{\bf
X}_{m}{\bf P}$, which leaves ${\cal H}$ invariant for the appropriate election
of $m$.

It follows that ${\bf Z}_{m}({\cal H})\subset{\cal H}$. Finally, $m$ is
disposed of: ${\bf Z}=m{\bf Z}_{m}$ is selfadjoint, satisfies ${\bf Z}{\bf
P}={\bf X}{\bf P}$, ${\bf Z}$ leaves ${\cal H}$ invariant, and

$1=\|{\bf Z}{\bf P}\|\leq\|{\bf Z}\|\leq 1,$

i.e. $\|{\bf Z}\|=1$. ∎

##  6 Minimality of curves in ${\cal I}_{A}^{a}$ with given initial conditions

In [2], a metric was introduced in the set of isometries of a Hilbert space.
If ${\cal I}$ denotes the set of isometries in ${\cal L}$, a natural metric in
${\cal I}$ is given by

$d(V,W)=\inf\\{\ell(\gamma):\gamma\hbox{ is a smooth curve in }{\cal I}\hbox{
joining }V\hbox{ and }W\\},$

where

$\ell(\gamma)=\int_{I}\|\dot{\gamma}(t)\|dt.$

Fix $V\in{\cal I}$. A vector ${\cal V}$ tangent to ${\cal I}$ at $V$ is the
velocity vector $\dot{\gamma}(0)$ of a curve $\gamma\subset{\cal I}$ with
$\gamma(0)=V$. A smooth curve $\gamma$ in ${\cal I}$ can be lifted to smooth
curve of unitary operators in ${\cal L}$: $\gamma(t)=\Gamma(t)V$ for $\Gamma$
in ${\cal U}({\cal L})$. It follows that ${\cal V}=iXV$, for $X^{*}=X\in{\cal
B}({\cal L})$. The main result established in [2] (Theorem 2.2) states the
following:

Suppose that $\|{\cal V}\|=1$, and let $P=VV^{*}$ be the final projection of
$V$. If $Z^{*}=Z$ satisfies that $\|Z\|=1$ and $ZP={\cal V}P$, then the curve

$\delta(t)=e^{itZ}V$

has minimal length along its path for $|t|\leq\pi$, among all smooth curves in
${\cal I}$ which join the same endpoints as $\delta$.

Define the metric in ${\cal I}_{A}^{a}$, by considering the norm of ${\cal
B}({\cal L})$ at every tangent space: if $T\in{\cal I}_{A}^{a}$ and ${\cal
V}\in(T{\cal I}_{A}^{a})_{T}$, then

$|{\cal V}|=\|\boldsymbol{{\cal V}}\|,$

where $\boldsymbol{{\cal V}}$ is the extension of ${\cal V}$ to ${\cal L}$.
Formally, since one has considered the regular structure of ${\cal I}_{A}^{a}$
regarding it as a submanifold of ${\cal B}_{A}({\cal H})$, one is introducing
a (constant) distribution of norms in the tangent bundle, consisting of the
norm defined by the $A$-inner product. Or, if one does not care about the
regular structure of ${\cal I}_{A}^{a}$, one just regards ${\cal I}_{A}^{a}$
as a subset of the metric space $({\cal I},d)$.

A direct consequence of the result cited above and Lemma 5.1 is the following:

######  Theorem 6.1.

Let $T\in{\cal I}^{a}_{A}$, and ${\cal V}\in(T{\cal I}_{A}^{a})_{T}$ with
$|{\cal V}|=1$. Then there exists a symmetrizable operator $Z\in{\cal
B}_{A}({\cal H})$ such that the curve

$\delta(t)=e^{it{\bf Z}}{\bf T}|_{\cal H}=e^{itZ}T\in{\cal I}_{A}^{a}$

satisfies that $\delta(0)=T$, $\dot{\delta}(0)={\cal V}$ and $\delta$ has
minimal length along its path for $|t|\leq\pi$, among all smooth curves in
${\cal I}^{a}_{A}$ which join the same endpoints as $\delta$. In fact,
$\delta$ is minimal also in the bigger manifold ${\cal I}$.

###### Proof.

The tangent vector ${\cal V}$ is of the form ${\cal V}=iXT$, for some
$X\in{\cal B}({\cal H})$ symmetrizable operator with $|{\cal V}|=\|{\bf X}{\bf
T}\|=1$. We claim that there exists ${\bf Z}^{*}={\bf Z}$ with $\|{\bf
Z}\|=1$, ${\bf Z}{\bf T}={\bf X}{\bf T}$ and ${\bf Z}({\cal H})\subset{\cal
H}$. Indeed, note that ${\bf Z}{\bf T}={\bf X}{\bf T}$ if and only if ${\bf
Z}{\bf P}={\bf X}{\bf P}$, for ${\bf P}$ the final projection of ${\bf T}$:

${\bf Z}{\bf T}={\bf X}{\bf T}\Longrightarrow{\bf Z}{\bf P}={\bf Z}{\bf T}{\bf
T}^{*}={\bf X}{\bf T}{\bf T}^{*}={\bf X}{\bf P}\Longrightarrow{\bf Z}{\bf
P}{\bf T}={\bf Z}{\bf T}={\bf X}{\bf P}{\bf T}={\bf X}{\bf P}.$

Also note that

$\|{\bf X}{\bf P}\|=\|{\bf X}{\bf T}{\bf T}^{*}\|\leq\|{\bf X}{\bf T}\|=\|{\bf
X}{\bf T}{\bf T}^{*}{\bf T}\|\leq\|{\bf X}{\bf T}{\bf T}^{*}\|=\|{\bf X}{\bf
P}\|,$

i.e. $\|{\bf X}{\bf P}\|=1$. If $Z={\bf Z}|_{\cal H}$, then
$\delta(t)=e^{itZ}T$ has minimal length along its path, due to Theorem 2.2 in
[2]. Clearly $\dot{\delta}(0)=ZT=XT={\cal V}$. ∎

##  7 Components of ${\cal I}_{A}^{a}$ and the action of the restricted group

The connected components of the set of usual isometries in a Hilbert space are
parametrized by the co-rank of the isometries: two isometries lie in the same
connected component (i.e., are conjugate by the left action of the unitary
group) if and only if they have the same co-rank $0\leq n\leq+\infty$. This
fact follows easily and is a consequence of the connectedness of the unitary
group of a Hilbert space. We do not know if the group ${\cal U}_{A}$ is
connected (or how the properties of the operator $A$ determine the components
of ${\cal U}_{A}$): we believe that it is an interesting problem. What does
hold in general, both for usual or $A$-isometries, is that two isometries are
conjugate by the group action if and only if their final projections are
conjugate. The argument is essentially contained in the discussion before
Theorem 4.3.

######  Proposition 7.1.

Let $T_{1},T_{2}\in{\cal I}_{A}^{a}$, with final $A$-orthogonal idempotents
$P_{1}=T_{1}T_{1}^{\sharp}$, $P_{2}=T_{2}T_{2}^{\sharp}$.

  1. 1.

There exists $G\in{\cal U}_{A}$ such that $GT_{1}=T_{2}$ if and only if there
exists $H\in{\cal U}_{A}$ such that $HP_{1}H^{-1}=P_{2}$.

  2. 2.

$T_{1}$ and $T_{2}$ lie in the same connected component of ${\cal I}_{A}^{a}$
if and only if $P_{1}$ and $P_{2}$ lie in the same component of ${\cal
P}_{A}$.

###### Proof.

If there exists $G\in{\cal I}_{A}^{a}$ such that $GT_{1}=T_{2}$, then

$P_{2}=T_{2}T_{2}^{\sharp}=GT_{1}(GT_{1})^{\sharp}=GT_{1}T_{1}^{\sharp}G^{\sharp}=GP_{1}G^{-1}.$

Conversely, if there exists $H\in{\cal U}_{A}$ such that $HP_{1}H^{-1}=P_{2}$,
then $T_{1}^{\prime}=HT_{1}$ and $T_{2}$ are $A$-isometries with the same
final space $P_{2}$. Consider the operator $K$ given by
$K=T_{2}(T_{1}^{\prime})^{\sharp}+1-P_{2}$. Note that

$KK^{\sharp}=(T_{2}(T_{1}^{\prime})^{\sharp}+1-P_{2})(T_{1}^{\prime}T_{2}^{\sharp}+1-P_{2})=T_{2}(T_{1}^{\prime})^{\sharp}T_{1}^{\prime}T_{2}^{\sharp}+1-P_{2},$

because
$T_{2}(T_{1}^{\prime})^{\sharp}(1-P_{2})=0=(1-P_{2})T_{1}^{\prime}T_{2}^{\sharp}$.
Since

$T_{2}(T_{1}^{\prime})^{\sharp}T_{1}^{\prime}T_{2}^{\sharp}=T_{2}T_{2}^{\sharp}=P_{2},$

then $KK^{\sharp}=1$. Similarly $K^{\sharp}K=1$, i.e. $K\in{\cal U}_{A}$.
Moreover

$KT_{1}^{\prime}=(T_{2}(T_{1}^{\prime})^{\sharp}+1-P_{2})T_{1}^{\prime}=T_{2}(T_{1}^{\prime})^{\sharp}T_{1}=T_{2}.$

Therefore, $T_{2}=KT_{1}^{\prime}=KHT_{1}$, with $KH\in{\cal U}_{A}$.

In order to prove part 2., one uses the fact that both

$\pi_{T_{1}}:{\cal U}_{A}\to\\{GT_{1}:G\in{\cal U}_{A}\\},\
\pi_{T_{1}}(G)=GT_{1}$

and

$\pi_{P_{1}}:{\cal U}_{A}\to\\{GP_{1}G^{-1}:G\in{\cal U}_{A}\\},\
\pi_{P_{1}}(G)=GP_{1}G^{-1}$

C∞ submersions, and that the orbits $\\{GT_{1}:G\in{\cal U}_{A}\\}$ and
$\\{GP_{1}G^{-1}:G\in{\cal U}_{A}\\}$ are unions of connected components of
${\cal I}_{A}^{a}$ and ${\cal P}_{A}$, respectively. Therefore a continuous
path $T(t)\in{\cal I}_{A}^{a}$ with $T(0)=T_{1}$ and $T(1)=T_{2}$ lifts to a
continuous path $G(t)\in{\cal U}_{A}$, i.e., $T(t)=G(t)T_{1}$. Then
$P(t)=G(t)P_{1}G^{-1}(t)$ is a continuous path in ${\cal P}_{A}$ joining
$P_{1}$ and $P_{2}$.

Similarly, if $P(t)$ is a continuous path in ${\cal P}_{A}$, there exists a
continuous path $H(t)\in{\cal U}_{A}$ such that $P(t)=H(t)P_{1}H^{-1}(t)$.
Then as in the proof of part 1.,

$K(t)=T_{2}(H(t)T_{1})^{\sharp}+1-P_{2}\in{\cal U}_{A}$

is a continuous path, and $T(t)=K(t)H(t)T_{1}$ is a continuous path in ${\cal
I}_{A}^{a}$, joining $T_{1}$ and $T_{2}$. ∎

In the remaining of this section, the action of the restricted group

${\cal U}_{A}^{\infty}:=\\{G\in{\cal U}_{A}:G-1\hbox{ is compact}\\}$

on ${\cal I}_{A}^{a}$ is considered. In [3] it was shown that ${\cal
U}_{A}^{\infty}$ is a $C^{\infty}$-Banach-Lie group, whose Banach-Lie group is

$\mathfrak{u}_{A}^{\infty}=\\{iX\in{\cal B}_{A}({\cal H}):X\hbox{ is
symmetrizable and compact}\\}.$

Denote by ${\cal K}({\cal H})$ the space of compact operators in ${\cal H}$.
In [3] it was also proved that the usual exponential map $\exp(iX)=e^{iX}$,

$\exp:\mathfrak{u}_{A}^{\infty}\to{\cal U}_{A}^{\infty}$

is surjective. This implies, in particular, that ${\cal U}_{A}^{\infty}$ is
connected. Let us consider the action of the restricted group ${\cal
U}_{A}^{\infty}$ on ${\cal I}_{A}^{a}$. The first result characterizes the
orbits of the restricted action. Note that these orbits are connected.

######  Theorem 7.2.

Let $T_{0}\in{\cal I}_{A}^{a}$. Then

$\\{GT_{0}:G\in{\cal U}_{A}^{\infty}\\}=\\{T\in{\cal
I}_{A}^{a}:T-T_{0}\in{\cal K}({\cal H})\\}.$

###### Proof.

If $T=GT_{0}$ for $G\in{\cal U}_{A}^{\infty}$, then clearly $T\in{\cal
I}_{A}^{a}$ and

$T-T_{0}=GT_{0}-T_{0}=(G-1)T_{0}\in{\cal K}({\cal H}).$

Conversely, suppose that $T\in{\cal I}_{A}^{a}$ satisfies that $T-T_{0}$ is
compact. Then, by a theorem of I.C. Gohberg and M.I. Zambickii [10] (see
Remark 2.2) the extension ${\bf T}-{\bf T}_{0}$ is compact in ${\cal L}$. Then
${\bf T}^{*}({\bf T}-{\bf T}_{0})=1-{\bf T}^{*}{\bf T}_{0}$ is compact, i.e.
${\bf T}^{*}{\bf T}_{0}$ is a Fredholm operator in ${\cal L}$ with zero index.
Then, since ${\bf T}$ is an isometric isomorphism between ${\cal L}$ and
$R({\bf T})=R({\bf T}{\bf T}^{*})$, and ${\bf T}_{0}^{*}$ is an isometric
isomorphism between $R({\bf T}_{0})=R({\bf T}_{0}{\bf T}_{0}^{*})$ and ${\cal
L}$, it follows that

${\bf T}{\bf T}^{*}{\bf T}_{0}{\bf T}_{0}^{*}:R({\bf T}_{0}{\bf T}_{0}^{*})\to
R({\bf T}{\bf T}^{*})$

is a Fredholm operator of zero index. In [4] (Theorem 6.3) it was shown this
implies that $P_{T}=TT^{\sharp}$ and $P_{T_{0}}=T_{0}T_{0}^{\sharp}$ lie in
the same connected component (same orbit) of the restricted Grassmannian:
there exists $G\in{\cal U}_{A}^{\infty}$ such that $GP_{T_{0}}G^{-1}=P_{T}$.
As in Proposition 7.1, let $K=T(GT_{0})^{\sharp}+1-P_{T}\in{\cal U}_{A}$,
which satisfies $KGT_{0}=T$. We claim that $K\in{\cal U}_{A}^{\infty}$.
Indeed,

$K-1=T(GT_{0})^{\sharp}-P_{T}=TT_{0}^{\sharp}G^{-1}-GT_{0}T_{0}^{\sharp}G^{-1}=(T-GT_{0})T_{0}^{\sharp}G^{-1},$

and since $G\in{\cal U}_{A}^{\infty}$ is of the form $G=1+C$ with $C\in{\cal
K}({\cal H})$,

$T-GT_{0}=T-(1+C)T_{0}=T-T_{0}-CT_{0}\in{\cal K}({\cal H}).$

∎

With a similar argument as the one used to prove that ${\cal I}_{A}^{a}$ has
differentiable structure, it can be shown that the orbit of $T_{0}$ under the
action on ${\cal U}_{A}^{\infty}$ is a $C^{\infty}$-manifold and a homogeneous
space of this group.

######  Proposition 7.3.

Given a fixed $T_{0}\in{\cal I}_{A}^{a}$, the set $\\{T\in{\cal
I}_{A}^{a}:T-T_{0}\in{\cal K}({\cal H})\\}$ is a complemented
$C^{\infty}$-submanifold of the affine space $T_{0}+{\cal B}_{A}({\cal
H})\cap{\cal K}({\cal H})$, and the onto map

$\pi^{\infty}_{T_{0}}:{\cal U}_{A}^{\infty}\to\\{T\in{\cal
I}_{A}^{a}:T-T_{0}\in{\cal K}({\cal H})\\},\ \pi^{\infty}_{T_{0}}(G)=GT_{0}$

is a $C^{\infty}$-submersion

###### Proof.

We shall use Lemma 4.2. Consider the Banach algebra ${\cal
B}=\mathbb{C}1+{\cal B}_{A}({\cal H})\cap{\cal K}({\cal H})$, with the norm
$|B|=\max\\{\|B\|,\|B\|^{\sharp}\\}$. The group ${\cal U}_{A}^{\infty}$ is a
subgroup of the Banach-Lie group of invertible operators in ${\cal B}$. The
element

$K=T(GT_{0})^{\sharp}+1-P_{T}\in{\cal U}_{A}^{\infty}$

defines a map - in fact a cross section - on a neighbourhood of $T_{0}$ in
$\\{T\in{\cal I}_{A}^{a}:T-T_{0}\in{\cal K}({\cal H})\\}$, with values in
${\cal U}_{A}^{\infty}$. Indeed, the operator $G$ is also a $C^{\infty}$ map
in the argument $T$ (because of the regular structure of the restricted orbit
of $P_{T_{0}}$ under the action of ${\cal U}_{A}^{\infty}$ [4]). Note that the
map $T\mapsto P_{T}=TT^{\sharp}$ is $C^{\infty}$, it is the restriction (to
the orbit of $T_{0}$) of the global $C^{\infty}$ map $B\mapsto BB^{\sharp}$ of
the ambient algebra ${\cal B}$. Clearly, then, the local cross section
$T\mapsto KG$ extends to a $C^{\infty}$ map defined on a neighbourhood of
$T_{0}$ in ${\cal B}$.

The space on which ${\cal U}_{A}^{\infty}$ acts by left multiplication, namely
$\\{X\in{\cal B}_{A}({\cal H}):X-T_{0}\in{\cal K}({\cal H})\\}$, is not a
Banach space. It is the affine Banach space $T_{0}+{\cal B}_{A}({\cal
H})\cap{\cal K}({\cal H})$. Therefore the argument proceeds, with slight
modifications. ∎

Aknowledgements

I wish to thank María Eugenia Di Iorio y Lucero for many helpful comments and
suggestions.

This work was supported by the grant PICT 2019 04060 (FONCyT - ANPCyT,
Argentina)

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# Coupling between conduction and near-field radiative heat transfer
in tip - plane geometry

Chams Gharib Ali Barura  Université Paris-Saclay, Institut d’Optique Graduate
School, CNRS, Laboratoire Charles Fabry, 91127, Palaiseau, France  Philippe
Ben-Abdallah  Université Paris-Saclay, Institut d’Optique Graduate School,
CNRS, Laboratoire Charles Fabry, 91127, Palaiseau, France  Riccardo Messina
<EMAIL_ADDRESS>Université Paris-Saclay, Institut
d’Optique Graduate School, CNRS, Laboratoire Charles Fabry, 91127, Palaiseau,
France

(August 27, 2024)

###### Abstract

We analyze the coupling between conduction and radiative heat transfer in
near-field regime between two coaxial cylinders separated by a vacuum gap. By
solving the heat transport equation in the steady-state regime between metals
or polar materials we highlight a flux saturation mechanism for the radiative
transfer even without non-local effect. In the case of polar materials this
saturation occurs in the separation distances range of $1$ to $10\,$nm which
can be experimentally explored.

Two bodies at different temperatures exchange energy even when separated by
vacuum via radiative heat transfer. The associated flux is limited in the far
field (for separation distances $d\gg\hbar c/k_{B}T$, of some microns at
ambient temperature) by Stefan - Boltzmann’s law. The advent of fluctuational
electrodynamics [1, 2] showed that this limit can be largerly overcome in the
near-field regime ($d\ll\hbar c/k_{B}T)$, where photons tunneling between the
two bodies can represent the main contribution to the energy exchange, in
particular when the two bodies support electromagnetic surface resonant modes,
such as surface waves [3] or a continuum of evanescent modes such as
hyperbolic modes [4]. This idea has triggered the discussion of a variety of
possible applications [5, 6, 7, 8, 11, 12, 9, 10, 13, 14, 15] and has been
explored experimentally in a number of experiments (see Refs. 16, 17, 18 for
some recent review papers), typically confirming the theoretical predictions.
Nevertheless, some of these experiments have observed deviations from the
results of fluctuational electrodynamics, both in the extreme-near-field
regime [19] (nanometer and sub-nanometer range of distances) and at tens of
nanometers [20], whose explanation remains to date elusive. Some theoretical
interpretations (mainly invoking the participation of different carriers to
the heat transfer) have been put forward in the extreme near field [21, 22,
23, 24, 25, 26, 27], but these can only apply below a few nanometers. Another
issue which has been investigated and could be relevant at slightly larger
distances is the coupling between conduction within each body and the
radiative flux exchanged between them. After some first works on this topic
[28, 29], it was shown [30] that for two parallel slabs in the diffusive
regime (thickness much larger than the mean free path of phonons) the
conduction - radiation coupling can induce a temperature profile within each
body along with a saturation of the exchanged flux. A more general study for
arbitrary transport regimes [31] has confirmed that this effect is strong in
the diffusive regime and can also strongly impact the thermalization dynamics
of close bodies [32]. The aim of the present work is to go beyond the
configuration of two parallel slabs and explore the impact of geometry on the
conduction - radiation coupling. More specifically, we are going to
investigate a geometry allowing us to simulate the experimentally interesting
configuration of a sharp tip in front of a planar substrate.

Figure 1: Sketch of the geometry of the system.

To start we consider two coaxial cylinders as sketched in (Fig. 1). The left
(right) one occupies the region $z_{1}<z<z_{2}$ ($z_{3}<z<z_{4}$) and have a
radius $R_{0}$ ($fR_{0}$, with $f\leq 1$). The two cylinders are aligned and
separated by a vacuum gap of thickness $d=z_{3}-z_{2}$. We assume that the
temperatures at the external boundaries $z=z_{1}$ and $z=z_{4}$ are kept
constant at $T_{L}$ and $T_{R}$, respectively, by two thermostats. Our purpose
is to solve the heat equation in steady state regime $\nabla(\kappa\nabla
T)=-\dot{Q}$, where $\kappa$ is the thermal conductivity of solids is assumed
to be the same in the two cylinders, $T$ is the temperature profile, and
$\dot{Q}$ a volumetric heat source stemming from the radiative heat transfer
between the two cylinders. Taking into account the fact that in the near field
the Poynting vector decays over a very thin layer of some nanometers inside
each body, we will assume, as done in Ref. 30, that the radiative heat
transfer can be modeled as a surface boundary condition, allowing us to reduce
the heat equation to a source-free equation. The solution of this equation
reads in cylindrical coordinates

$T(r,z)=a_{0}+b_{0}z+\sum_{k=1}^{\infty}\Bigl{(}a_{k}e^{\alpha_{k}\frac{z}{R}}+b_{k}e^{-\alpha_{k}\frac{z}{R}}\Bigr{)}J_{0}\Bigl{(}\alpha_{k}\frac{r}{R}\Bigr{)},$
(1)

where $J_{n}(x)$ is the $n$-th order Bessel function of the first kind,
$\\{\alpha_{i}\\}$ ($i=0,1,\dots$) is the set of zeros of $J_{1}(x)$ and the
coefficients $\\{a_{i},b_{i}\\}$ for $i=0,1,\dots$ have to be determined. The
externally imposed temperatures translate into $T(r,z_{1})=T_{L}$ and
$T(r,z_{4})=T_{R}$, whereas in the internal boundaries $z=z_{2},z_{3}$ we
impose the continuity of the flux $\partial T(r,z_{i})/\partial
r=-\varphi_{i}(r)/\kappa$, where $i=2,3$ and $\varphi_{i}(r)$ represents the
surface energy flux on the surface at $z_{i}$. At this stage, we add a further
approximation, inspired by the Derjaguin approximation [36], widely used when
modeling near-field radiative heat transfer in complex geometries. We assume
that heat is exchanged only between points of two surfaces $z=z_{2},z_{3}$
having the same $r$ coordinate. This implies that the surface flux
$\varphi_{2}(r)$ at $z=z_{2}$ vanishes for $r>fR_{0}$, while for $r\leq
fR_{0}$ we have $\varphi_{2}(r)=\varphi_{3}(r)=\varphi(r)$. For the expression
of $\varphi(r)$, we employ the calculation of the flux between two planar
parallel slabs. [33, 34] $\Phi(T(r,z_{2}),T(r,z_{3}),d)$ at the local
temperatures $T(r,z_{2})$ and $T(r,z_{3})$ and placed at distance $d$ between
each other.

We first assume that the flux takes the form
$\varphi(r)=\Phi(T(r=0,z_{2}),T(r=0,z_{3}),d)$, i.e. only depends on the
temperatures at the center of the two inferfaces facing each other. This is
equivalent to neglecting any $r$ dependence of the temperature in the right
cylinder. This allows us to solve the coupled heat equations analytically,
obtaining for the smaller cylinder ($z_{3}<z<z_{4}$) the linear profile

$T(r,z)=T_{R}+\frac{\Phi(T(0,z_{2}),T(0,z_{3}),d)}{\kappa}(z_{4}-z),$ (2)

and for the larger one ($z_{1}<z<z_{2}$)

$\begin{split}T(r,z)&=T_{L}-f^{2}\frac{\Phi(T(0,z_{2}),T(0,z_{3}),d)}{\kappa}(z-z_{1})\\\
&\,-2fR_{0}\frac{\Phi(T(0,z_{2}),T(0,z_{3}),d)}{\kappa}\\\
&\,\times\sum_{k=1}^{\infty}\frac{J_{1}(f\alpha_{k})}{\alpha_{k}^{2}J_{0}^{2}(\alpha_{k})}\frac{\sinh\Bigl{[}\frac{\alpha_{k}(z-z_{1})}{R_{0}}\Bigr{]}}{\cosh\Bigl{(}\frac{\alpha_{k}\delta_{L}}{R_{0}}\Bigr{)}}J_{0}\Bigl{(}\alpha_{k}\frac{r}{R}\Bigr{)}.\end{split}$
(3)

Note that this solution is still implicit, since the right-hand sides of both
Eqs. 2 and 3 depend on $T(0,z_{2})$ and $T(0,z_{3})$. A fully closed-form
solution can be obtained by introducing, as already done in Ref. 30, the
following explicit form of the exchanged flux

$\Phi(T(0,z_{2}),T(0,z_{3}),d)\simeq\frac{\gamma[T(0,z_{2})-T(0,z_{3})]}{d^{\beta}},$
(4)

where we linearized the flux with respect to the temperature difference and
have assumed a power law scaling $d^{-\beta}$ with respect to the distance
[3]. For polar materials in slab - slab configuration we have $\beta=2$. By
evaluating Eqs. 2 and 3 at $r=0$ and $z=z_{3},z_{2}$, respectively, by
subtracting them and coupling this to Eq. 4 we obtain the full closed-form
analytical expression of the temperature profile in the two cylinders

$T(r,z)=\begin{cases}T_{L}-\frac{\gamma(T_{L}-T_{R})}{\xi}\Biggl{[}f^{2}(z-z_{1})+2R_{0}f\sum_{k=1}^{\infty}\frac{J_{1}(f\alpha_{k})}{\alpha_{k}^{2}J_{0}^{2}(\alpha_{k})}\frac{\sinh\Bigl{[}\frac{\alpha_{k}(z-z_{1})}{R_{0}}\Bigr{]}}{\cosh\Bigl{[}\frac{\alpha_{k}(z_{2}-z_{1})}{R_{0}}\Bigr{]}}J_{0}\Bigl{(}\alpha_{k}\frac{r}{R}\Bigr{)}\Biggr{]},&z_{1}<z<z_{2},\\\
T_{R}+\frac{\gamma(T_{L}-T_{R})}{\xi}(z_{4}-z),&z_{3}<z<z_{4},\end{cases}$ (5)

where

$\begin{split}\xi&=\kappa d^{2}+\gamma(f^{2}\delta_{L}+\delta_{R})+2\gamma
R_{0}f\Gamma\Bigl{(}f,\frac{\delta_{L}}{R_{0}}\Bigr{)},\\\
\Gamma(f,\beta)&=\sum_{k=1}^{\infty}J_{1}(f\alpha_{k})\tanh(\alpha_{k}\beta)/[\alpha_{k}^{2}J_{0}^{2}(\alpha_{k})],\end{split}$
(6)

along with the one of the exchanged flux

$\varphi(d,f,R_{0})=\frac{\frac{\gamma\kappa(T_{L}-T_{R})}{\kappa
d^{2}}}{1+\frac{\gamma}{\kappa
d^{2}}\Bigl{[}f^{2}\delta_{L}+\delta_{R}+2R_{0}f\Gamma\Bigl{(}f,\frac{\delta_{L}}{R_{0}}\Bigr{)}\Bigr{]}},$
(7)

$\delta_{L}=z_{2}-z_{1}$ ($\delta_{R}=z_{4}-z_{3}$) being the height of the
larger (smaller) cylinder. Note that Eq. (7) allows to recover the diverging
flux $\Phi(d)=\gamma(T_{L}-T_{R})/d^{2}$ in the absence of conduction -
radiation coupling, corresponding to $\kappa\to\infty$, and the results for
two parallel slabs [30] for $f=1$ [note that $\Gamma(1,x)=0$].

Figure 2 shows the radiative heat flux for $(T_{L},T_{R})=(400,300)\,$K (fixed
throughout the paper) for two polar materials, silicon dioxide (SiO2) and
silicon carbide (SiC). Optical data are taken from Ref. 35, the conductivities
are $\kappa_{\text{SiO}_{2}}=1.4\,$W/$\text{m}\cdot\text{K}$ and
$\kappa_{\text{SiC}}=120\,$W/$\text{m}\cdot\text{K}$, and the values of
$\gamma$ (see Eq. (4)) stem from a conventional fluctuational-electrodynamics
calculation, yielding $\gamma_{\text{SiO}_{2}}=3.8\times 10^{-12}\,$W/K and
$\gamma_{\text{SiC}}=1.3\times 10^{-12}\,$W/K. We have taken $R_{0}=10\,\mu$m
and $f=10^{-2}$, resulting in a tip radius $fR_{0}=100\,$nm. The solid black
and red lines (SiO2 and SiC, respectively) clearly highlight a saturation
effect already observed for two parallel slabs [30] ($f=1$, dot-dashed lines
in Fig. 2). Compared to this configuration, while the deviation with respect
to the theoretical $d^{-2}$ divergence (absence of coupling) takes place in
the same distance range, the flux saturates to higher values in the case of
two cylinders, in agreement with the lower flux exchanged through radiation.
Moreover, we observe that the flux is higher for SiO2 with respect to SiC in
the absence of coupling (dashed lines), while the opposite is true in the
presence of coupling, both in a tip - plane and a slab - slab configurations.
This is a consequence of a stronger conduction - radiation coupling for SiO2,
stemming from a higher radiative flux
($\gamma_{\text{SiO}_{2}}>\gamma_{\text{SiC}}$) and a lower thermal
conductivity ($\kappa_{\text{SiO}_{2}}<\kappa_{\text{SiC}}$). More
quantitatively, while the coupling should be observable in the nanometer range
for SiC, this distance range moves to tens of nanometers for SiO2. An estimate
of the characteristic distance at which this deviation becomes relevant is
given by $\tilde{d}=\sqrt{2\gamma\delta/\kappa}$, at which for two slabs the
flux is halved with respect to the absence of coupling
($\varphi(\tilde{d},f=1,R_{0})=\Phi(d)/2$). This gives
$\tilde{d}_{\text{SiO}_{2}}=23$ nm for SiO2 and $\tilde{d}_{\text{SiC}}=1.5$
nm for SiC, in agreement with the numerical results.

Figure 2: Radiative heat flux as a function of $d$ for $R_{0}=10\,\mu$m and
$\delta_{L}=\delta_{R}=100\,\mu$m. The solid line corresponds to a tip - plane
configuration ($f=10^{-2}$), the dot-dashed line to the slab-slab scenario
($f=1$), whereas the dashed line accounts for the $d^{-2}$ divergence (absence
of coupling). The different colors correspond to different materials: black
for SiO2, red for SiC and blue for Au, for which the effect of coupling is not
observable.

We have also performed our analysis by using the full expression of the
radiative heat flux at distance $d$ stemming from fluctuational
electrodynamics, i.e by abandoning both the linearization with respect to
$T(0,z_{2})-T(0,z_{3})$ and the $d^{-2}$ scaling of the flux. By evaluating
Eqs. (2) and (3) at $r=0$ and $z=z_{3},z_{2}$, respectively, and by injecting
these expressions in $\Phi(T(0,z_{2}),T(0,z_{3}),d)$, we easily obtain the
non-linear equation $\Phi(T_{L}-ax,T_{R}+bx)-x=0$, where $a$ and $b$ can be
easily obtained from Eqs. (2) and (3), while $x$ represents the flux
$\varphi(T(0,z_{2}),T(0,z_{3}),d)$, from which the temperature profile in both
cylinders can be deduced. The solution of this equation can be easily found
numerically, yielding for any separation distance $d$ the full temperature
profiles and exchanged flux in the presence of coupling. This approach allowed
us to verify the validity of Eq. (4), obtaining curves which are
indistinguishable from those shown in Fig. 2 for polar materials. Moreover, it
allowed us to investigate the role of coupling in the case of metals, for
which the $d^{-2}$ scaling law for the distance dependence of flux is not
valid anymore. More specifically, we have considered two gold (Au) cylinders
whose dielectric constant is described by means of a Drude model [35] and
having a thermal conductivity
$\kappa_{\text{Au}}=310\,$W/$\text{m}\cdot\text{K}$. Figure 2 shows that the
curves for gold in the absence of coupling, and in the presence of coupling
both for two slabs and in the tip - plane configuration are superposed. This
is, not surprisingly, a result of the high thermal conductivity of gold along
with the much lower radiative flux due to the absence of surface resonant
modes in the infrared region of the spectrum.

Finally, we have verified the validity of the approximation neglecting the $r$
dependence of the flux, by replacing Eq. (4) by the expression
$\varphi(r)\simeq\gamma[T(r,z_{2})-T(r,z_{3})]/d^{2}$. By solving the time-
independent heat equation in the presence of this $r$-dependent flux and
injecting the solution of the two temperatures inside its expression we obtain
the integral equation for the exchanged flux

$\begin{split}&\varphi(r)=\frac{\gamma(T_{L}-T_{R})}{d^{2}}-\frac{2\gamma}{\kappa
d^{2}}\Biggl{\\{}\frac{f^{2}\delta_{L}+\delta_{R}}{f^{2}R_{0}^{2}}\int_{0}^{fR_{0}}\\!dr^{\prime}r^{\prime}\,\varphi(r^{\prime})+\sum_{k=1}^{\infty}\frac{1}{\alpha_{k}fR_{0}J_{0}^{2}(\alpha_{k})}\\\
&\times\int_{0}^{fR_{0}}\\!dr^{\prime}r^{\prime}\,\Bigl{[}f\tanh\Bigl{[}\frac{\alpha_{k}\delta_{L}}{R_{0}}\Bigr{]}J_{0}\Bigl{(}\frac{\alpha_{k}r}{R_{0}}\Bigr{)}J_{0}\Bigl{(}\alpha_{k}\frac{r^{\prime}}{R_{0}}\Bigr{)}+\tanh\Bigl{[}\frac{\alpha_{k}\delta_{R}}{fR_{0}}\Bigr{]}J_{0}\Bigl{(}\alpha_{k}\frac{r}{fR_{0}}\Bigr{)}J_{0}\Bigl{(}\frac{\alpha_{k}r^{\prime}}{fR_{0}}\Bigr{)}\Bigr{]}\varphi(r^{\prime})\Biggr{\\}}.\end{split}$
(8)

For any experimentally reasonable choice of the parameters $d$, $R_{0}$ and
$f$, the numerical solutions of this equation both for SiC and SiO2 are
indistinguishable from the ones of Fig. 2, confirming the negligible
dependence of the temperature profiles on the radial coordinate.

We have used the formalism developed so far to investigate the role of
geometry in the conduction - radiation coupling. For two SiO2 cylinders at a
distance $d=1\,$nm, Fig. 3 shows the ratio between the exchanged flux in the
presence of coupling $\varphi(d,f,R_{0})$ and the one in the absence of
coupling $\Phi(d)$, as a function of the radius $R_{0}$ of the larger cylinder
(representing the plane) and the ratio $f$ between the two radii.

Figure 3: Ratio between the exchanged fluxes in the presence and absence of
coupling as a function of the radius $R_{0}$ of the larger cylinder and the
ratio $f$ between the two cylinders radii. The other parameters are $d=1\,$nm
and $\delta_{L}=\delta_{R}=100\,\mu$m. Some contour lines are indicated in
red, while the black dashed line corresponds to a fixed tip radius
$fR_{0}=100\,$nm.

It also highlight a reduction of the flux between two and three orders of
magnitude. Moreover, albeit the large range of explored parameters, this ratio
is not significantly varying, proving that we are already in a saturation
regime. Furthermore, it is also interesting to look at a cut of this density
plot for a given value of the tip radius $fR_{0}$. A value of $fR_{0}=100\,$nm
is shown in Fig. 3, allowing us to conclude that not only is this saturation
clearly taking place for small tips, but that for those values we are safely
in a saturation regime.

We conclude by studying the temperature $T(r=0,z=z_{3})$ of the apex of tip.
This quantity is shown in Fig. 4 as a function of $f$ for different radii
$R_{0}$ of the larger cylinder. We identify two limiting cases: for $f=1$ we
have two parallel slabs both thermalizing, in their closest points
$z=z_{2},z_{3}$, to $350\,$K, i.e. the average of the two externally imposed
temperatures $T_{L}$ and $T_{R}$. On the contrary, for $f\to 0$ the apex of
the tip tends to thermalize to the temperature of the left cylinder (the
plane), almost independent of $z$ and equal to $T_{L}$. These limiting
temperatures strongly depend on the material under scrutiny (and on the
distance). They can be expressed analytically, within the approximations
leading to Eq. (5), as
$T_{f=1}=T_{R}+\gamma(T_{L}-T_{R})\delta_{R}/[\gamma(\delta_{L}+\delta_{R})+\kappa^{2}d]$
and $T_{f\to
0}=T_{R}+\gamma(T_{L}-T_{R})\delta_{R}/(\gamma\delta_{R}+\kappa^{2}d)$ for
$f=1$ and $f\to 0$, respectively. For SiC this gives $T_{f\to 0}=364$ K and
$T_{f=1}=335$ K , showing a present but weaker conduction - radiation coupling
with respect to SiO2, while in the case of gold we obtain $T(r=0,z_{3})\simeq
T_{f\to 0}\simeq T_{f=1}\simeq T_{R}=300\,$K confirming the negligible
conduction - radiation coupling.

Figure 4: Temperature of the apex of the tip as a function of the radii
fraction $f$, for three different values of the radius $R_{0}$ of the larger
cylinder. The other parameters are $d=1\,$nm and
$\delta_{L}=\delta_{R}=100\,\mu$m.

Going back to SiO2 (see Fig. 4), we observe that the transition between the
limiting cases $f\to 0$ and $f=1$ takes place on a characteristic scale of $f$
which clearly depends on $R_{0}$. Nevertheless, while the three curves depict
the possible transitions from the extremely thin tip to the slab - slab
configuration for different values of $R_{0}$, it must be pointed out that for
any reasonable value of the tip radius $fR_{0}$ we are always very close to
the limit $f\to 0$, i.e. fully in a saturated regime where the tip apex
thermalizes to the highest possible temperature.

In conclusion, we have studied the coupling between conductive and near-field
radiative heat transfer in a system made of two different coaxial cylinders
which mimic, in the limit of small ratio between the two cylinder radii, the
geometric configuration of a tip - plane thermal-microscopy experiment. We
have solved the coupled heat equation in this geometry, obtaining some closed-
form expressions for both the temperature profiles and the exchanged flux in
some approximations, whose validity has been verified numerically. We have
shown that the conduction-radiation coupling can induce a saturation of heat
flux exchanged between the two solids even without non-local optical response
of materials. This saturation can induce a significant temperature change in
the smaller cylinder (the tip), up to tens of degrees in the case of polar
materials, along with a strong reduction of the flux (of more than two orders
of magnitude) with respect to the scenario of absence of coupling. While this
coupling has a negligible effect in the case of metals, it is relevant for
polar materials and manifests itself at separation distances up to tens of
nanometers, making its experimental observability feasible. Since this
distance is comparable to distances where the non-local optical response of
materials is likely to play a role in the transfer, these effects should also
be analyzed in future works.

## Author declarations

### Conflict of interest

The authors have no conflicts to disclose.

## Data availability

The data that support the findings of this study are available from the
corresponding authors upon reasonable request.

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