$\mathbb{A}=\begin{pmatrix}a_{1}&a_{2}&b_{1}&b_{2}&b_{3}\\\
a_{3}&a_{4}&b_{4}&b_{5}&b_{6}\\\ a_{5}&a_{6}&c_{1}&c_{2}&d_{1}\\\
a_{7}&a_{8}&c_{3}&c_{4}&d_{2}\\\ a_{9}&a_{10}&c_{5}&c_{6}&d_{3}\end{pmatrix}$

partitioned into four obvious blocks $A,B,C,D$.

If $\mathbb{A}$ is partitioned into $r$ blocks
$A_{k}\in\mathbb{M}_{n_{k},m_{k}}$, we write

$\mathbb{A}=\bigcup_{k=1}^{r}A_{k}=A_{1}\cup\cdots\cup A_{r}$ (8.1.2)

where we can use the $=$ sign if one considers $A_{k}$ not only as an element
of $\mathbb{M}_{n_{k},m_{k}}$ but also as a submatrix of $\mathbb{A}$ with its
position in $\mathbb{A}$.

We say that the partitioning (8.1.2) is column compatible, or that
$\mathbb{A}$ is partitioned into column compatible blocks if for all pairs of
indexes $k,l$, either $A_{k}$ and $A_{l}$ lie on the same set of columns of
$\mathbb{A}$, or $A_{k}$ and $A_{l}$ lie on two disjoint sets of columns of
$\mathbb{A}$. Similarly, (8.1.2) is row compatible, if for all pairs of
indexes $k,l$, either $A_{k}$ and $A_{l}$ lie on the same set of rows of
$\mathbb{A}$, or$A_{k}$ and $A_{l}$ lie on two disjoint sets of rows of
$\mathbb{A}$.

Our Pythagorean Theorem 8.2.1 will be stated for row or column compatible
blocks. An application is a Theorem of Bhatia and Kittaneh for the Schatten
$p$-norms (Corollary 8.2.5). Another application is an inequality for the
singular values of compression onto hyperplanes. A matrix $A\in\mathbb{M}_{d}$
is an operator on $\mathbb{C}^{d}$. Given a hyperplane ${\mathcal{S}}$ of
$\mathbb{C}^{d}$, we have a unit vector $h$ such that
${\mathcal{S}}=h^{\perp}$, that is $x\in{\mathcal{S}}\iff\langle
h,x\rangle=h^{*}x=0$. The compression $A_{\mathcal{S}}$ of $A$ onto
${\mathcal{S}}$ is the operator acting on ${\mathcal{S}}$ defined as the
restriction of $EA$ to ${\mathcal{S}}$ where $E$ stands for the (orthogonal)
projection onto ${\mathcal{S}}$. Theorem 8.2.1 entails a bound for the
singular values of $A_{\mathcal{S}}$ in terms of those of $A$. These results
are given in Section 3; we state a special case in the following corollary.
Let $\mu_{j}$ denote the $j$-th singular value arranged in nonincreasing
order.

######  Corollary 8.1.1.

Let $A\in\mathbb{M}_{d}$ be a normal matrix and let ${\mathcal{S}}$ be a
hyperplane of $\mathbb{C}^{d}$ orthogonal to the unit vector $h$. Set
$\beta=\|Ah\|^{2}-|\langle h,Ah\rangle|^{2}$. Then, for $j=1,\ldots,d-1$,

$\mu_{j}^{2}(A)\geq\mu_{j}^{2}(A_{\mathcal{S}})\geq\mu^{2}_{j+1}(A)-\beta.$

We discuss the case of four and five blocks in Section 4. For four blocks, our
Pythagorean theorem entails an interesting inequality stated in the next
corollary.

######  Corollary 8.1.2.

Let $\mathbb{A}\in\mathbb{M}_{d,d^{\prime}}$ be partitioned into four blocks
$A,B,C,D$. Then, there exist some isometries $U,V,W,X$ of suitable sizes such
that

$2|\mathbb{A}|\geq U|A|U^{*}+V|B|V^{*}+W|C|W^{*}+X|D|X^{*}.$

The last section is devoted to several other operator inequalities such as the
first inequality in the abstract.

####  8.2 A Pythagorean theorem

######  Theorem 8.2.1.

Let $\mathbb{A}\in\mathbb{M}_{d,d^{\prime}}$ be partitioned into $r$ row or
column compatible blocks $A_{k}\in\mathbb{M}_{n_{k},m_{k}}$. Then, there exist
some isometries $U_{k}\in\mathbb{M}_{d^{\prime},m_{k}}$ such that

$|\mathbb{A}|^{2}=\sum_{k=1}^{r}U_{k}|A_{k}|^{2}U_{k}^{*}.$

Recall that $U\in\mathbb{M}_{d^{\prime},m}$, $m\leq d^{\prime}$, is an
isometry if $U^{*}U=\mathbb{1}_{m}$, the identity on $\mathbb{C}^{m}$. If
$\mathbb{A}\in\mathbb{M}_{d,1}$, then the theorem reads as Pythagoras’
Theorem.

###### Proof.

Consider a positive matrix in $\mathbb{M}_{n+m}$ partitioned as

$\begin{bmatrix}A&X\\\ X^{*}&B\end{bmatrix}$

with diagonal blocks $A\in\mathbb{M}_{n}$ and $B\in\mathbb{M}_{m}$. By [31,
Lemma 3.4] we have two unitary matrices $U,V\in\mathbb{M}_{n+m}$ such that

$\begin{bmatrix}A&X\\\ X^{*}&B\end{bmatrix}=U\begin{bmatrix}A&0\\\
0&0\end{bmatrix}U^{*}+V\begin{bmatrix}0&0\\\ 0&B\end{bmatrix}V^{*},$ (8.2.1)

equivalently,

$\begin{bmatrix}A&X\\\ X^{*}&B\end{bmatrix}=U_{1}AU_{1}^{*}+U_{2}BU_{2}^{*}$

for two isometry matrices $U_{1}\in\mathbb{M}_{n+m,n}$ and
$U_{2}\in\mathbb{M}_{n+m,m}$. An obvious iteration of (8.2.1) shows that,
given a positive block matrix in $\mathbb{M}_{m}$ partitioned into $p\times p$
blocks,

$\mathbb{B}=\left(B_{i,j}\right)_{1\leq i,j\leq p},$

with square diagonal blocks $B_{i,i}\in\mathbb{M}_{n_{i}}$ and
$n_{1}+\cdots+n_{p}=m$, we have the decomposition

$\mathbb{B}=\sum_{i=1}^{p}U_{i}B_{i,i}U_{i}^{*}$ (8.2.2)

for some isometries $U_{i}\in\mathbb{M}_{m,n_{i}}$.

We use (8.2.2) to prove the theorem. Consider first the column compatible
case. Thus we have a partitioning into $p$ block columns,

$\mathbb{A}={\mathbf{C}}_{1}\cup\cdots\cup{\mathbf{C}}_{p},$ (8.2.3)

and each block $A_{k}$ belongs to one block column ${\mathbf{C}}_{q}$. By
relabelling the $A_{k}$’s if necessary, we may assume that we have $p$
integers $1=\alpha_{1}<\alpha_{2}<\cdots<\alpha_{p}<r$ such that

${\mathbf{C}}_{q}=A_{\alpha_{q}}\cup\cdots\cup A_{\alpha_{q+1}-1},\quad 1\leq
q<p,\quad{\text{and}}\quad{\mathbf{C}}_{p}=A_{\alpha_{p}}\cup\cdots\cup
A_{\alpha_{r}}.$

We also have a partitioning into $p$ block rows,

$\mathbb{A}^{*}={\mathbf{C}}^{*}_{1}\cup\cdots\cup{\mathbf{C}}^{*}_{p},$
(8.2.4)

and multiplying (8.2.4) and (8.2.3) we then obtain a block matrix for
$\mathbb{A}^{*}\mathbb{A}=|\mathbb{A}|^{2}\in\mathbb{M}_{d^{\prime}}$,

$|\mathbb{A}|^{2}=\left({\mathbf{C}}^{*}_{i}{\mathbf{C}}_{j}\right)_{1\leq
i,j\leq p}.$

By the decomposition (8.2.2) we have

$|\mathbb{A}|^{2}=\sum_{i=1}^{p}U_{i}{\mathbf{C}}^{*}_{i}{\mathbf{C}}_{i}U_{i}^{*}$

for some isometries $U_{i}\in\mathbb{M}_{d^{\prime},n_{i}}$, where $n_{i}$ is
the number of columns of ${\mathbf{C}}_{i}$. Hence, with the convention
$\alpha_{p+1}:=r+1$,

$|\mathbb{A}|^{2}=\sum_{i=1}^{p}\sum_{k=\alpha_{i}}^{\alpha_{i+1}-1}U_{i}A_{k}^{*}A_{k}U_{i}^{*}$

establishing the theorem for a column compatible partitioning.

Now, we turn to the row compatible case. Thus we have a partitioning into $p$
block rows,

$\mathbb{A}={\mathbf{R}}_{1}\cup\cdots\cup{\mathbf{R}}_{p},$ (8.2.5)

and each block $A_{k}$ belongs to one block row ${\mathbf{R}}_{q}$ and, as in
the column compatible case, we may assume that we have $p$ integers
$1=\alpha_{1}<\alpha_{2}<\cdots<\alpha_{p}<r$ such that

${\mathbf{R}}_{q}=A_{\alpha_{q}}\cup\cdots\cup A_{\alpha_{q+1}-1},\quad 1\leq
q<p,\quad{\text{and}}\quad{\mathbf{R}}_{p}=A_{\alpha_{p}}\cup\cdots\cup
A_{\alpha_{r}}.$

We also have a partitioning into $p$ block columns,

$\mathbb{A}^{*}={\mathbf{R}}^{*}_{1}\cup\cdots\cup{\mathbf{R}}^{*}_{p}$
(8.2.6)

Mutiply (8.2.6) and (8.2.5) and note that

$|\mathbb{A}|^{2}=\sum_{l=1}^{p}{\mathbf{R}}^{*}_{l}{\mathbf{R}}_{l}.$ (8.2.7)

with $p$ block matrices in $\mathbb{M}_{d^{\prime}}$, ($l=1,\ldots,p$),

${\mathbf{R}}^{*}_{l}{\mathbf{R}}_{l}=\left(A^{*}_{i}A_{j}\right)_{\alpha_{l}\leq
i,j<\alpha_{l+1}}$ (8.2.8)

where we still use $\alpha_{p+1}:=r+1$. Applying the decomposition (8.2.2) to
the block matrices (8.2.8) yields

${\mathbf{R}}^{*}_{l}{\mathbf{R}}_{l}=\sum_{i=\alpha_{l}}^{\alpha_{l+1}-1}U_{i}|A_{i}|^{2}U_{i}^{*}$

for some isometries $U_{i}$ of suitable sizes, and combining with (8.2.7)
completes the proof. ∎

Denote by $\mu_{1}(S)\geq\mu_{2}(S)\geq\cdots$ the singular values of a matrix
$S\in\mathbb{M}_{n,m}$. This list is often limited to $\min\\{n,m\\}$
elements, however we can naturally define $\mu_{k}(S)=0$ for any index $k$
larger than $\min\\{n,m\\}$. Given two matrices of same size, a classical
inequality of Weyl asserts that

$\mu_{j+k+1}(S+T)\leq\mu_{j+1}(S)+\mu_{k+1}(T)$

for all nonnegative integers $j$ and $k$. This inequality and Theorem 8.2.1
entail the next corollary.

######  Corollary 8.2.2.

Let $\mathbb{A}\in\mathbb{M}_{d,d^{\prime}}$ be partitioned into $r$ row or
column compatible blocks $A_{k}\in\mathbb{M}_{n_{k},m_{k}}$. Then, for all
nonnegative integers $j_{1},j_{2},\ldots,j_{r}$,

$\mu^{2}_{j_{1}+j_{2}+\cdots+j_{r}+1}(\mathbb{A})\leq\sum_{k=1}^{r}\mu^{2}_{j_{k}+1}(A_{k}).$

A special case of this inequality is given in the abstract for three blocks
and $j_{1}=2$, $j_{2}=1$, $j_{3}=0$.

Since any partitioning into three blocks is row or column compatible we have
the next corollary.

######  Corollary 8.2.3.

Let $\mathbb{A}\in\mathbb{M}_{d,d^{\prime}}$ be partitioned into three blocks
$A,B,C$. Then, there exist some isometries $U,V,W$ of suitable sizes such that

$|\mathbb{A}|^{2}=U|A|^{2}U^{*}+V|B|^{2}V^{*}+W|C|^{2}W^{*}.$

By using the triangle inequality for the Schatten $p$-norms we have the trace
inequality

$\left\\{{\mathrm{Tr}}\,|\mathbb{A}|^{2p}\right\\}^{1/p}\leq\left\\{{\mathrm{Tr}}\,|A|^{2p}\right\\}^{1/p}+\left\\{{\mathrm{Tr}}\,|B|^{2p}\right\\}^{1/p}+\left\\{{\mathrm{Tr}}\,|C|^{2p}\right\\}^{1/p},\quad
p\geq 1,$ (8.2.9)

equivalently

$\|\mathbb{A}\|^{2}_{q}\leq\|A\|^{2}_{q}+\|B\|^{2}_{q}+\|C\|^{2}_{q}$ (8.2.10)

for all Schatten $q$-norms, $q\geq 2$.

Theorem 8.2.1 entails another interesting relation between the blocks of a
partitioned matrix and the full matrix.

######  Corollary 8.2.4.

Let $\mathbb{A}\in\mathbb{M}_{d,d^{\prime}}$ be partitioned into $r$ row or
column compatible blocks $A_{k}\in\mathbb{M}_{n_{k},m_{k}}$. Then, for some
isometries $V_{j}\in\mathbb{M}_{m,d^{\prime}}$, with $m=\sum_{k=1}^{r}m_{k}$,

$\bigoplus_{k=1}^{r}|A_{k}|^{2}=\frac{1}{r}\sum_{j=1}^{r}V_{j}|\mathbb{A}|^{2}V_{j}^{*}.$

###### Proof.

From Theorem 8.2.1 and the main result of [35] we have

$\bigoplus_{k=1}^{r}U_{k}|A_{k}|^{2}U_{k}=\frac{1}{r}\sum_{j=1}^{r}W_{j}|\mathbb{A}|^{2}W_{j}^{*}$

for some isometries $U_{k}\in\mathbb{M}_{d^{\prime},m_{k}}$ and some
isometries $W_{j}\in\mathbb{M}_{rd^{\prime},d^{\prime}}$. Since

$\bigoplus_{k=1}^{r}|A_{k}|^{2}=C\left\\{\bigoplus_{k=1}^{r}U_{k}|A_{k}|^{2}U_{k}\right\\}C^{*}$

for some contraction $C\in\mathbb{M}_{m,rd^{\prime}}$, we infer

$\bigoplus_{k=1}^{r}|A_{k}|^{2}=\frac{1}{r}\sum_{j=1}^{r}CW_{j}|\mathbb{A}|^{2}W_{j}^{*}C^{*}.$

If $|\mathbb{A}|$ is invertible, then, taking trace, the above equality
ensures that contractions $CW_{j}$ satisfy
$W_{j}^{*}C^{*}CW_{j}={\mathbf{1}}_{d^{\prime}}$ for all $j$. Hence the result
is proved with $V_{j}=CW_{j}$. The general case follows by a limit argument. ∎

We are in a position to estimate the Schatten norms of the blocks with the
full matrix. The following corollary was first obtained by Bhatia and Kittaneh
[17] in case of a matrix partitioned into $n\times n$ blocks.

######  Corollary 8.2.5.

Let $\mathbb{A}\in\mathbb{M}_{d,d^{\prime}}$ be partitioned into $r$ row or
column compatible blocks $A_{k}\in\mathbb{M}_{n_{k},m_{k}}$. Then, for all
Schatten $q$-norms, $q\geq 2$,

$r^{{\frac{2}{q}}-1}\sum_{k=1}^{r}\|A_{k}\|_{q}^{2}\leq\|\mathbb{A}\|_{q}^{2}\leq\sum_{k=1}^{r}\|A_{k}\|_{q}^{2}$

These two inequalities are reversed for $2>q>0$.

###### Proof.

For $p:=q/2\geq 1$, the second inequality contains (8.2.10) and immediately
follows from Theorem 8.2.1 and the triangle inequality for the Schatten
$p$-norms. Corollary 8.2.4 gives

$\||\mathbb{A}|^{2}\|_{p}\geq\left\||A_{1}|^{2}\oplus\cdots\oplus|A_{r}|^{2}\right\|_{p}$

and since the concavity of $t\mapsto t^{1/p}$ entails

$\left\||A_{1}|^{2}\oplus\cdots\oplus|A_{r}|^{2}\right\|_{p}=\left(\||A_{1}|^{2}\|_{p}^{p}+\cdots\||A_{r}|^{2}\|_{p}^{p}\right)^{1/p}\geq
r^{\frac{1}{p}-1}\left(\||A_{1}|^{2}\|_{p}+\cdots+\||A_{r}|^{2}\|_{p}\right)$

we get the first inequality. These inequalities are reversed for $0<p<1$. ∎

Corollary 8.2.4 is relevant to Majorisation Theory. We take this opportunity
to point out an interesting fact about majorisation in the next proposition.
Though this result might be well-known to some experts, it does not seem to be
in the literature. Let $A,B\in\mathbb{M}_{n}^{+}$, the positive semi-definite
cone of $\mathbb{M}_{n}$. The majorisation $A\prec B$ means that

$\sum_{j=1}^{k}\mu_{j}(A)\leq\sum_{j=1}^{k}\mu_{j}(B)$

for all $k=1,2,\ldots n$, with equality for $k=n$. The majorisation $A\prec B$
is equivalent to

$A=\sum_{i=1}^{n}\alpha_{i}U_{i}BU_{i}^{*}$

for some unitary matrices $U_{i}\in\mathbb{M}_{n}$ and weights $\alpha_{i}\geq
0$ with $\sum_{i=1}^{n}\alpha_{i}=1$. This can be easily derived from
Caratheodory’s theorem [94]. A more accurate statement holds.

######  Proposition 8.2.6.

Let $A,B\in\mathbb{M}_{n}^{+}$, $A\prec B$.Then, for some unitary matrices
$U_{i}\in\mathbb{M}_{n}$,

$A=\frac{1}{n}\sum_{i=1}^{n}U_{i}BU_{i}^{*}.$

###### Proof.

By the Schur-Horn Theorem, we may assume that $A$ is the diagonal part of $B$.
Then we use the simple idea of Equation (2) in the nice paper of Bhatia [14].
∎

Note that Corollary 8.2.4 may be restated as

$\bigoplus_{k=1}^{r}|A_{k}|^{2}=\frac{1}{r}\sum_{j=1}^{r}U_{j}(|\mathbb{A}|^{2}\oplus
O)U_{j}^{*}.$

for some unitary matrices $U_{j}$ and some fixed zero matrix $O$. Hence we
have an average of $r$ matrices in the unitary orbit of
$|\mathbb{A}|^{2}\oplus O$, this number $r$ being (much) smaller than the one
given by Proposition 8.2.6, $d=m_{1}+\cdots+m_{r}$.

####  8.3 Compression onto a hyperplane

By a hyperplane of $\mathbb{C}^{d}$ we mean a vector subspace of dimension
$d-1$. The next corollary is a singular value version of Cauchy’s Interlacing
Theorem [13, p. 59].

######  Corollary 8.3.1.

Let $A\in\mathbb{M}_{d}$ and let ${\mathcal{S}}$ be a hyperplane of
$\mathbb{C}^{d}$ orthogonal to the unit vector $h$. Set
$\beta=\min\\{\|Ah\|^{2},\|A^{*}h\|^{2}\\}-|\langle h,Ah\rangle|^{2}$. Then,
for all $j=1,\ldots,d-1$,

$\mu_{j}^{2}(A)\geq\mu_{j}^{2}(A_{\mathcal{S}})\geq\mu_{j+1}^{2}(A)-\beta.$

This double inequality is stronger than
$\mu_{j}(A)\geq\mu_{j}(A_{\mathcal{S}})\geq\mu_{j+1}(A)-\sqrt{\beta}$. If $A$
is a normal matrix, then $\|Ah\|=\|A^{*}h\|$ and we have Corollary 8.1.1. If
$A=V$ is a unitary matrix, $\mu_{j}(V)=1$ for all $j$ and
$\|Vh\|=\|V^{*}h\|=1$ for all unit vectors, so we deduce from Corollary 8.3.1
that $\mu_{j}(V_{\mathcal{S}})\geq|\langle h,Vh\rangle|$. In fact one can
easily check that $\mu_{j}(V_{\mathcal{S}})=1$ for $j\leq d-2$ and
$\mu_{d-1}(V_{\mathcal{S}})=|\langle h,Vh\rangle|$. Hence Corollary 8.3.1 is
sharp.

###### Proof.

(Corollary 8.3.1) The inequality $\mu_{j}(A)\geq\mu_{j}(A_{\mathcal{S}})$ is
trivial. To deal with the other inequality we may assume that $h$ is the last
vector of the canonical basis and that $A_{\mathcal{S}}$ is the submatrix of
$A$ obtained by deleting the last column and the last line. We partition $A$
as

$A=A_{\mathcal{S}}\cup B\cup C$

where $B$ contains the $d-1$ entries below $A_{\mathcal{S}}$ and $C$ is the
last column of $A$. We then apply to this partitioning Corollary 8.2.2 with
$j_{1}=j-1$, $j_{2}=0$, and $j_{3}=1$ to get

$\mu_{j+1}^{2}(A)\leq\mu_{j}^{2}(A_{\mathcal{S}})+\mu_{1}^{2}(B)+\mu_{2}^{2}(C).$

Since $\mu_{2}(C)=0$, we have

$\ \mu_{j+1}^{2}(A)-\mu_{1}^{2}(B)\leq\mu_{j}^{2}(A_{\mathcal{S}})$

Observe that $\mu_{1}^{2}(B)=\|A^{*}h\|^{2}-|\langle h,Ah\rangle|^{2}$, hence

$\mu_{j}^{2}(A_{\mathcal{S}})\geq\mu_{j+1}^{2}(A)-\|A^{*}h\|^{2}+|\langle
h,Ah\rangle|^{2}.$ (8.3.1)

We may also partition $A$ as

$A=A_{\mathcal{S}}\cup R\cup L$

where $R$ contains the $d-1$ entries on the right of $A_{\mathcal{S}}$ and $L$
stands for the last line of $A$. Arguing as above with $R$ and $L$ in place of
$B$ and $C$ yields

$\mu_{j+1}^{2}(A)-\mu_{j}^{2}(A_{\mathcal{S}})\leq\mu_{1}^{2}(R)=\|Ah\|^{2}-|\langle
h,Ah\rangle|^{2}$ (8.3.2)

Combining (8.3.1) and (8.3.2) completes the proof. ∎

######  Corollary 8.3.2.

Let $\mathbb{A}\in\mathbb{M}_{d,d^{\prime}}$ be partitioned into $r$ row or
column compatible blocks $A_{k}\in\mathbb{M}_{n_{k},m_{k}}$. Then, for each
block $A_{k}$ and all $j\geq 1$,

$\mu^{2}_{j}(\mathbb{A})-\mu^{2}_{j}(A_{k})\leq\sum_{l\neq
k}\mu^{2}_{1}(A_{l}).$

###### Proof.

Apply Corollary 8.2.2 with $j_{k}=j-1$ and $j_{l}=0$ for all $l\neq k$. ∎

######  Corollary 8.3.3.

Let $A\in\mathbb{M}_{d}$ and let ${\mathcal{S}}$ be a hyperplane of
$\mathbb{C}^{d}$ orthogonal to the unit vector $h$. Then for all
$j=1,\ldots,d-1$,

$\mu^{2}_{j}(A)-\mu^{2}_{j}(A_{\mathcal{S}})\leq\|Ah\|^{2}+\|A^{*}h\|^{2}-|\langle
h,Ah\rangle|^{2}.$

###### Proof.

We may suppose that $h$ is the last vector of the canonical basis and we
partition $A$ into three blocks : $A_{\mathcal{S}}$, the last column of $A$,
and the $d-1$ entries below $A_{\mathcal{S}}$. We then apply the previous
corollary. ∎

####  8.4 Four and five blocks

Partitionings into four blocks are not necessarily row or column compatible.
However, for such partitionings, Theorem 8.2.1 still holds.

######  Corollary 8.4.1.

Let $\mathbb{A}\in\mathbb{M}_{d,d^{\prime}}$ be partitioned into four blocks
$A,B,C,D$. Then, there exist some isometries $U,V,W,X$ of suitable sizes such
that

$|\mathbb{A}|^{2}=U|A|^{2}U^{*}+V|B|^{2}V^{*}+W|C|^{2}W^{*}+X|D|^{2}X^{*}.$

###### Proof.

We assume that $A$ is the block in the upper left corner and we distinguish
three cases.

(1) $A$ has the same number $d$ of lines as $\mathbb{A}$. In such a case,
letting $A^{\prime}=B\cup C\cup D$, the partitioning $\mathbb{A}=A\cup
A^{\prime}$ is column compatible, and we have two isometry matrices
$U,U^{\prime}$ such that

$|\mathbb{A}|^{2}=U|A|^{2}U^{*}+U^{\prime}|A^{\prime}|^{2}U^{\prime*}.$
(8.4.1)

Since $A^{\prime}$ is partitioned into three blocks, necessarily a row or
column partitioning, we can apply the theorem to obtain the decomposition

$|A^{\prime}|^{2}=V^{\prime}|B|^{2}V^{\prime*}+W^{\prime}|B|^{2}W^{\prime*}+X^{\prime}|B|^{2}X^{\prime*}$
(8.4.2)

for some isometry matrices $V^{\prime},W^{\prime},X^{\prime}$ of suitable
sizes. Combining (8.4.1) and (8.4.2) we get the conclusion of the corollary
with the isometry matrices $V=U^{\prime}V^{\prime}$, $W=U^{\prime}W^{\prime}$,
and $X=U^{\prime}X^{\prime}$.

(2) $A$ has the same number $d^{\prime}$ of columns as $\mathbb{A}$. Letting
again $A^{\prime}=B\cup C\cup D$, the partitioning $\mathbb{A}=A\cup
A^{\prime}$ is row compatible, and we may argue as in case (1).

(3) $A$ has $l<d$ lines and $c<d^{\prime}$ columns. There exist then a block,
say $B$, on the top position, and just on the right of $A$, and another block,
say $C$ just below $A$ and on the left side. We consider three subcases (a),
(b), (c).

(a) $B$ has fewer than $l$ lines. Then, the last block $D$ is necessarily
below $B$ with the same number of columns as $B$, and so $C$ has either the
same number of columns as $A$ or $C$ has $d^{\prime}$ columns as $\mathbb{A}$.
In the first case, $\mathbb{A}=A\cup C\cup B\cup D$ is a column compatible
partitioning and we can apply the theorem. In the second case, the situation
is the same as in (2).

(b) $B$ has exactly $l$ lines, like $A$. We denote by $\gamma$ the number of
columns of $B$ and we consider three situations.

(i) $C$ has more than $c+\gamma$ columns. Then necessarily $C$ has
$d^{\prime}$ columns and $D$ is the upper right block with $l$ lines, hence
$\mathbb{A}=A\cup B\cup D\cup C$ is a line compatible partitioning and we may
apply the theorem.

(ii) $C$ has exactly $c+\gamma$ columns. Then, letting
$\mathbb{A}^{\prime\prime}=A\cup B\cup C$ with have a partitioning into three
blocks, and $\mathbb{A}=\mathbb{A}^{\prime\prime}\cup D$. Thus applying the
theorem twice as in case (1) yields the conclusion.

(iii) $C$ has fewer than $c+\gamma$ columns. Then $D$ is the lower right
block, with the same number of lines as $B$, and $\mathbb{A}$ is partitioned
into line compatible blocks. Thus the theorem can be applied.

(c) $B$ has more lines than $A$. Let $\lambda$ be the number of line of $B$.
Hence $\lambda>l$. There exist two situations

(iv) $\lambda<d$. Then $D$ is the lower right block, with the same number of
columns as $B$, and $\mathbb{A}$ is partitioned into line compatible blocks.
Thus we may apply the theorem.

(v) $\lambda=d$. Then $A^{\prime\prime\prime}=A\cup C\cup D$ is a partitioning
into three blocks and $\mathbb{A}=A^{\prime\prime\prime}\cup B$, thus applying
twice the theorem completes the proof. ∎

We do not know whether Corollary 8.4.1 can be extended or not to any
partitioning in five blocks. For instance we are not able to prove or disprove
a version of Corollary 8.4.1 for the matrices

$\mathbb{A}=\begin{pmatrix}a_{1}&a_{2}&b_{1}\\\ d_{1}&x&b_{2}\\\
d_{2}&c_{1}&c_{2}\\\
\end{pmatrix}\quad{\mathrm{or}}\quad\mathbb{A}=\begin{pmatrix}a_{1}&a_{2}&a_{3}&b_{1}&b_{2}\\\
a_{4}&a_{5}&a_{6}&b_{3}&b_{4}\\\ d_{1}&d_{2}&x&b_{5}&b_{6}\\\
d_{3}&d_{4}&c_{1}&c_{2}&c_{3}\\\ d_{5}&d_{6}&c_{4}&c_{5}&c_{6}\end{pmatrix}$

partitioned into five obvious blocks $A,B,C,D,X$. Hence, that Theorem 8.2.1
holds or not for any partitioning into five blocks is an open problem. More
generally, we may consider the following two questions.

######  Question 8.4.2.

For which partitionings does Theorem 8.2.1 hold ? For which partitionings does
Corollary 8.2.2 hold ?

Matrices partitioned into four blocks (usually of same size) are comon
examples of partitionings. A nontrivial inequality follows from the previous
corollary.

######  Corollary 8.4.3.

Let $\mathbb{A}\in\mathbb{M}_{d,d^{\prime}}$ be partitioned into four blocks
$A,B,C,D$, and let $p>2$. Then, there exist some isometries $U,V,W,X$ of
suitable sizes such that

$2^{2-p}|\mathbb{A}|^{p}\leq
U|A|^{p}U^{*}+V|B|^{p}V^{*}+W|C|^{p}W^{*}+X|D|^{p}X^{*}.$

The inequality reverses for $2>p>0$.

Letting $p=1$ we have Corollary 8.1.2 with the constant 2 which is sharp, even
for a positive block matrix, as shown by the simple example

$\mathbb{A}=\begin{bmatrix}A&A\\\ A&A\end{bmatrix}.$

###### Proof.

For any monotone convex function $f(t)$ on the nonnegative axis, we have
thanks to [31, Corollary 2.4] and Corollary 8.4.1,

$\displaystyle f\left(\frac{|\mathbb{A}|^{2}}{4}\right)$
$\displaystyle=f\left(\frac{U|A|^{2}U^{*}+V|B|^{2}V^{*}+W|C|^{2}W^{*}+X|D|^{2}X^{*}}{4}\right)$
$\displaystyle\leq\Lambda\frac{f(U|A|^{2}U^{*})+V(f|B|^{2}V^{*})+f(W|C|^{2}W^{*})+f(X|D|^{2}X^{*})}{4}\Lambda^{*}$

for some unitary matrix $\Lambda\in\mathbb{M}_{d}^{\prime}$. Picking
$f(t)=t^{p/2}$ with $p>2$ yields the result. The reverse inequalities hold for
monotone concave functions $f(t)$ and $0<p<2$. ∎

######  Remark 8.4.4.

The version of Corollary 8.4.3 for three blocks $A,B,C$, and $p=1$ reads as
the inequality of the abstract,

$\sqrt{3}|\mathbb{A}|\geq U|A|U^{*}+V|B|V^{*}+W|C|W^{*}.$

The constant $\sqrt{3}$ is the best one: we cannot take a smaller constant for

$\mathbb{A}=\begin{bmatrix}x&y&z\\\ x&y&z\\\ x&y&z\end{bmatrix}$

partitioned into its three lines. For two blocks, a similar sharp inequality
holds with the constant $\sqrt{2}$.

####  8.5 Concave or convex functions

For sake of simplicity we state our results for a square matrix $\mathbb{A}$
partitioned into blocks. By adding some zero rows or zero columns to a
rectangular matrix, we could obtain statements for rectangular matrices
(Remark 8.5.9).

Suppose that $\mathbb{A}\in\mathbb{M}_{d}$ is partitioned into blocks
$A_{k}\in\mathbb{M}_{n_{k},m_{k}}$, $k=1,\ldots,r$. From Thompson’s triangle
inequality ([92] or [13, p. 74] we have

$|\mathbb{A}|\leq\sum_{k=1}^{r}U_{k}|A_{k}|U_{k}^{*}$ (8.5.1)

for some isometry matrices $U_{k}\in\mathbb{M}_{d,m_{k}}$. The equality of
Theorem 8.2.1 and (8.5.1) suggest several other inequalities, in particular,
if $\mathbb{A}$ is partioned in row or column compatible blocks,

$|\mathbb{A}|^{3}\geq\sum_{k=1}^{r}V_{k}|A_{k}|^{3}V_{k}^{*}$ (8.5.2)

for some isometries $V_{k}\in\mathbb{M}_{d,m_{k}}$. This is indeed true as
shown in the following theorem. We do not know if (8.5.2) can be extended to
any partitioning. Corollary 8.4.1 and the proof of Theorem 8.5.1 show that
(8.5.2) holds for four blocks. The case of five blocks is open.

######  Theorem 8.5.1.

Let $\mathbb{A}\in\mathbb{M}_{d}$ be partitioned into $r$ row or column
compatible blocks $A_{k}\in\mathbb{M}_{n_{k},m_{k}}$, and let $\psi(t)$ be a
monotone function on $[0,\infty)$ such that $\psi(\sqrt{t})$ is convex and
$\psi(0)=0$. Then there exist some isometries $V_{k}\in\mathbb{M}_{d,m_{k}}$
such that

$\psi(|\mathbb{A}|)\geq\sum_{k=1}^{r}V_{k}\psi(|A_{k}|)V_{k}^{*}.$

Theorem 8.5.1 considerably improves (8.2.9). A special case with
$\psi(t)=t^{3}$ is given in the abstract.

###### Proof.

Let $g(t)$ be a monotone convex function on $[0,\infty)$ such that $g(0)\leq
0$, and let $A,B\in\mathbb{M}_{n}$ be positive (semidefinite). By [12] or [31,
Corollary 3.2] we have

$g(A+B)\geq Ug(A)U^{*}+Vg(B)V^{*}$

for some unitary matrices $U,V\in\mathbb{M}_{n}$. Using this inequality and
Theorem 8.2.1 we infer

$g(|\mathbb{A}|^{2})\geq\sum_{k=1}^{r}W_{k}g(U_{k}|A_{k}|^{2}U_{k}^{*})W_{k}^{*}$

for some unitary matrices $W_{k}$ and some isometry matrices
$U_{k}\in\mathbb{M}_{d,m_{k}}$. If $g(0)=0$, we have
$g(U_{k}|A_{k}|^{2}U_{k}^{*})=U_{k}g(|A_{k}|^{2})U_{k}^{*}$. Hence

$g(|\mathbb{A}|^{2})\geq\sum_{k=1}^{r}V_{k}g(|A_{k}|^{2})V_{k}^{*}$

with the isometry matrices $V_{k}=W_{k}U_{k}$. Applying this to
$g(t)=\psi(\sqrt{t})$ completes the proof. ∎

######  Corollary 8.5.2.

Let $\mathbb{A}\in\mathbb{M}_{d}$ be partitioned into $r$ row or column
compatible blocks $A_{k}\in\mathbb{M}_{n_{k},m_{k}}$, and let $\varphi(t)$ be
a nonnegative function on $[0,\infty)$ such that $\varphi(\sqrt{t})$ is
concave. Then there exist some isometries $U_{k}\in\mathbb{M}_{d,n_{k}}$ such
that

$\varphi(|\mathbb{A}|)\leq\sum_{k=1}^{r}U_{k}\varphi(|A_{k}|)U_{k}^{*}.$

###### Proof.

Since $\varphi(\sqrt{t})$ is nonnegative and concave, it is necessarily a
monotone function (nondecreasing), hence continuous on $(0,\infty)$. Since we
are dealing with matrices we may further suppose that $\varphi(t)$ is also
continuous at $t=0$.

(1) Assume that $\varphi(0)=0$. Theorem 8.5.1 applied to $\psi(t)=-\varphi(t)$
proves the corollary.

(2) Assume that $\varphi(0)>0$. Since the continuous functional calculus is
continuous on the positive semidefinite cone of any $\mathbb{M}_{m}$, by a
limit argument, we may assume that $|\mathbb{A}|$ is invertible. So, suppose
that the spectrum of $|\mathbb{A}|^{2}$ lies in an interval $[r^{2},s^{2}]$
with $r>0$. Define a convex function $\phi(\sqrt{t})$ by
$\phi(\sqrt{t})=\varphi(\sqrt{t})$ for $t\geq r^{2}$, $\phi(0)=0$, and the
graph of $\phi(\sqrt{t})$ on $[0,r^{2}]$ is a line segment. Hence
$\phi(t)\leq\varphi(t)$ and $\phi(|\mathbb{A}|)=\varphi(|\mathbb{A}|)$.
Applying case (1) to $\phi$ yields

$\varphi(|\mathbb{A}|)=\phi(|\mathbb{A}|)\leq\sum_{k=1}^{r}U_{k}\phi(|A_{k}|)U_{k}^{*}\leq\sum_{k=1}^{r}U_{k}\varphi(|A_{k}|)U_{k}^{*}$

for some isometry matrices $U_{k}$. ∎

The next three corollaries follow from Corollary 8.5.2.

######  Corollary 8.5.3.

Let $\mathbb{A}\in\mathbb{M}_{mn}$ be partitioned into an $m\times m$ family
of blocks $A_{i,j}\in\mathbb{M}_{n}$, and let $0<q\leq 2$. Then there exist
some isometries $U_{i,j}\in\mathbb{M}_{mn,n}$ such that

$|\mathbb{A}|^{q}\leq\sum_{i,j=1}^{m}U_{i,j}|A_{i,j}|^{q}U_{i,j}^{*}.$

######  Corollary 8.5.4.

Let $\mathbb{A}\in\mathbb{M}_{mn}$ be partitioned into an $m\times m$ family
of blocks $A_{i,j}\in\mathbb{M}_{n}$, let $s\geq 1$ and $0<q\leq 2$. Then,

$\left\\{{\mathrm{Tr}\,}|\mathbb{A}|^{qs}\right\\}^{1/s}\leq\sum_{i,j=1}^{m}\left\\{{\mathrm{Tr}\,}|A_{i,j}|^{qs}\right\\}^{1/s}.$

######  Corollary 8.5.5.

Let $A\in\mathbb{M}_{n}$, let $c_{k}$ be the norm of the $k$-th column of $A$
and let $0<q\leq 2$. Then there exist some rank one projections
$E_{k}\in\mathbb{M}_{n}$ such that

$|\mathbb{A}|^{q}\leq\sum_{k=1}^{n}c_{k}^{q}E_{k}$

The last corollaries follow from Theorem 8.5.1.

######  Corollary 8.5.6.

Let $\mathbb{A}\in\mathbb{M}_{mn}$ be partitioned into an $m\times m$ family
of blocks $A_{i,j}\in\mathbb{M}_{n}$, and let $p\geq 2$. Then there exist some
isometries $U_{i,j}\in\mathbb{M}_{mn,n}$ such that

$|\mathbb{A}|^{p}\geq\sum_{i,j=1}^{m}U_{i,j}|A_{i,j}|^{p}U_{i,j}^{*}.$

######  Corollary 8.5.7.

Let $\mathbb{A}\in\mathbb{M}_{mn}$ be partitioned into an $m\times m$ family
of blocks $A_{i,j}\in\mathbb{M}_{n}$, let $0\leq s\leq 1$ and $p\geq 2$. Then,

$\left\\{{\mathrm{Tr}\,}|\mathbb{A}|^{ps}\right\\}^{1/s}\geq\sum_{i,j=1}^{m}\left\\{{\mathrm{Tr}\,}|A_{i,j}|^{ps}\right\\}^{1/s}.$

######  Corollary 8.5.8.

Let $A\in\mathbb{M}_{n}$, let $r_{k}$ be the norm of the $k$-th row of $A$ and
let $p\geq 2$. Then there exist some rank one projections
$E_{k}\in\mathbb{M}_{n}$ such that

$|\mathbb{A}|^{p}\geq\sum_{k=1}^{n}c_{k}^{p}E_{k}.$

######  Remark 8.5.9.

The proof of Theorem 8.5.1 is the same for a $d\times d^{\prime}$ matrix
$\mathbb{A}$. So Corollary 8.5.2 also holds for
$\mathbb{A}\in\mathbb{M}_{d,d^{\prime}}$ if $\varphi(0)=0$. In case of $d\geq
d^{\prime}$, we may again use a limit argument and assume that $|\mathbb{A}|$
is invertible. In case of $d^{\prime}>d$ we may argue as follows. Add some
zero lines to $\mathbb{A}$ in order to obtain a square matrix
$\mathbb{A}_{0}\in\mathbb{M}_{d}^{\prime}$. Let $B_{1}\ldots,B_{p}$ be the
blocks at the bottom of $\mathbb{A}$, and $R_{1},\ldots,R_{q}$ be the
remaining blocks of $\mathbb{A}$. Add some zeros to the blocks $B_{i}$ in
order to obtain blocks $B_{i}^{0}$ of $\mathbb{A}_{0}$ in such a way that

$\mathbb{A}_{0}=\left(\bigcup_{i}B_{i}^{0}\right)\cup\left(\bigcup_{j}R_{j}\right)$

is a row or column compatible partitioning of $\mathbb{A}_{0}$. Since it is a
square matrix, we may apply Corollary 8.5.2 and since
$|\mathbb{A}_{0}|=|\mathbb{A}|$ and $|B_{i}^{0}|=|B^{i}|$, we see that
Corollary 8.5.2 holds for $d\times d^{\prime}$ matrices.

####  8.6 Around this article

For a positive block-diagonal matrix, Theorem 8.2.1 is trivial. The following
statement [35] is more interesting.

######  Theorem 8.6.1.

Let $A_{i}\in\mathbb{M}_{n}^{+}$, $i\in\mathbb{I}_{m}$. Then, for some
isometries $V_{k}\in\mathbb{M}_{mn,n}$, $k\in\mathbb{I}_{m}$,

$\bigoplus_{i=1}^{m}A_{i}=\frac{1}{m}\sum_{k=1}^{m}V_{k}\left\\{\sum_{i=1}^{m}A_{i}\right\\}V_{k}^{*}.$

This says that direct sums are averages of usual sums, up to isometric
congruences. It is a genuine non-commutative fact with no analogous statement
for positive vectors and permutations of their components.

####  8.7 References of Chapter 8

  * [12

] J.S. Aujla and J.-C. Bourin, Eigenvalue inequalities for convex and log-
convex functions, Linear Algebra Appl. 424 (2007), 25–35.

  * [13

] R. Bhatia, Matrix Analysis, Gradutate Texts in Mathematics, Springer, New-
York, 1996.

  * [14

] R. Bhatia, Pinching, trimming, truncating, and averaging of matrices. Amer.
Math. Monthly 107 (2000),

  * [17

] R. Bhatia and F. Kittaneh, Norm inequalities for partitioned operators and
an application. Math. Ann. 287 (1990), no. 4, 719–726. no. 7, 602–608.

  * [31

] J.-C. Bourin and E.-Y. Lee, Unitary orbits of Hermitian operators with
convex or concave functions, Bull. Lond. Math. Soc. 44 (2012), no. 6,
1085–1102.

  * [35

] J.-C. Bourin and E.-Y. Lee, Direct sums of positive semi-definite matrices.
Linear Algebra Appl. 463 (2014), 273–281.

  * [42

] J.-C. Bourin and E.-Y. Lee, A Pythagorean Theorem for partitioned matrices,
Proc. Amer. Math. Soc., in press.

  * [92

] R.-C. Thompson, Convex and concave functions of singular values of matrix
sums, Pacific J. Math. 66 (1976), 285–290.

  * [94

] X. Zhan, The sharp Rado theorem for majorizations, Amer. Math. Monthly 110
(2003) 152–153.

###  Chapter 9 The spectral Theorem

####  9.1 Introduction

Hermitian (or symmetric) matrices acting on ${\mathcal{H}}_{n}=\mathbb{C}^{n}$
(or $\mathbb{R}^{n}$) form a nice finite dimensional real vector space, let us
denote it by $\mathbb{S}({\mathcal{H}}_{n})$, with a order structure, $A\leq
B$, whenever $B-A$ is positive semidefinte.

The fundamental property of $\mathbb{S}({\mathcal{H}}_{n}$) is the matrix
spectral theorem asserting that a Hermitian matrix can be diagonalized in a
suitable orthonormal basis. It is a very important theorem and, important too,
it is an easy result well understood by students. Combined with the order
structure, this theorem shows that Hermitian matrices can be regarded as a
generalization of $\mathbb{R}^{n}$ and opens the way of a full theory of
Matrix Analysis, where facts for real finite sequences of numbers search for
counterparts in the Hermitian matrix world.

But, the extension of this diagonalization theorem for operators on a infinite
dimensional separable Hilbert space ${\mathcal{H}}$ \- a key stone result of
Functional Analysis - is more delicate: the standard literature invoke
abstract constructions based on the Gelfand isomorphism between some abstract
algebras. It might be a too abstract and unnatural approach; the link with
matrices unexpectedly disappears though operators obviously have matrix
representations.

However, and fortunately as it is conceptually desirable, it is possible, to
give a simple proof deriving from the matrix case. Hence operators are not
more complicated than matrices any longer ! We do this rather easy and
definitely pleasant job in this note.

####  9.2 The matrix case, and two natural definitions for operators

Let us first recall a sketch of the proof of the spectral theorem for
Hermitian matrices.

  * •

if a subspace ${\mathcal{S}}\subset{\mathcal{H}}_{n}$ is invariant by $A$,
then so is its orthocomplement ${\mathcal{S}}^{\perp}$.

  * •

$\lambda_{1}^{\uparrow}(A):=\min\langle h,Ah\rangle$ must be an eigenvalue,

so that repeating the process with the restriction of $A$ to the
orthocomplement …. This is the variational method. and we arrive to the
following spectral decomposition of $A$:

$A=\sum_{j=1}^{k}\lambda_{j}^{\uparrow}(A)P_{j}$ (9.2.1)

A very important feature of this decompostion is to allow a matrix functional
calculus, given $f(t)\in C^{0}$, the matrix $f(A)$ makes sense. This is the
starting point of a branch of Matrix analysis. Note that the matrix functional
calculus behave well with the order structure of $C^{0}$,: $f(t)\geq g(t))$
implies $f(A)\geq g(A)$ for all Hermitian matrices $A$. this will be crucial
in our approach.

Now, let us consider the typical situation in the infinite dimensional case,
the multiplication operator on $L_{2}([0,1],$

$Z:f(t)\mapsto tf(t),\qquad f(t)\in L_{2}([0,1],{\mathrm{d}}t)$

This basic example shows that it is problematic to use eigenvalues and
projection $P_{j}$ as in (9.2.1). However it seems possible to have a
corresponding notion for the sum

$E_{j}=P_{1}+P_{2}+\cdots P_{j},\qquad j=1,2\cdots$ (9.2.2)

of the first $j$ spectral projections of $A$ by using the operators

$E_{w}:f(t)\mapsto{\mathbb{I}}_{[0,w]}(t)f(t),\qquad w\in[0,1].$

This is a very good point, because we may write the matrix spectral theorem
(9.2.1) as:

$A=\lambda_{1}^{\uparrow}(A)E_{1}+\lambda_{2}^{\uparrow}(A)(E_{2}-E_{1})+\lambda_{3}^{\uparrow}(A)(E_{3}-E_{2})+....+\lambda_{n}^{\uparrow}(A)(E_{n}-E_{n-1})$
(9.2.3)

We will obtain a similar statement for Hilbert space operators. To adapt the
matrix proof to the infinite dimensional setting, we need two simple
definitions. Let $\mathbb{S}({\mathcal{H}})$ be the set of Hermitian operators
on a infinite dimensional, separable Hilbert space ${\mathcal{H}}$.

######  Definition 9.2.1.

(sot-convergence, or strong convergence) We say that the sequence
$\\{X_{n}\\}$ in $\mathbb{S}({\mathcal{H}})$ is sot convergent, or strongly
converges, if it is norm bounded and, for all vectors $h\in{\mathcal{H}}$
(equivalently for all elements of an orthonormal basis), $\\{X_{n}h\\}$
converges in ${\mathcal{H}})$. If $\\{X_{n}\\}$ is sot convergent, then there
exists (a unic) $X\in\mathbb{S}({\mathcal{H}})$ such that $Xh=\lim_{n}X_{n}h$
for all vectors $h\in{\mathcal{H}}$ and we say that $X$ is the strong limit of
${X_{n}}$, written as $X={\mathrm{sot}}\lim_{n\to\infty}X_{n}$.

Hence the strong convergence is merely the pointwise convergence in
$\mathbb{S}({\mathcal{H}})$. Of course, it is a classical notion and ”sot”
refers to strong operator topology.

Given $A\in\mathbb{S}({\mathcal{H}})$, and fixing a (orthonormal) basis, we
may write $A$ as an infinite matrix $A=(a_{i,j})$, and so extracting the
$n$-by-$n$ left upper corners, $n=1,2,3,....$ we obtain a basic sequence of
Hermitian matrices for which the matrix spectral theorem and the functional
calculus are available. Since we wish to extend it to $A$, it seems natural to
use this sequence of matrices, and the following terminology is convenient.

######  Definition 9.2.2.

The sequence $\\{A_{n}\\}$ in $\mathbb{S}({\mathcal{H}})$ is a basic sequence
for $A$ if there exists an increasing sequence of finite rank projections
$\\{E_{n}\\}$, sot-convergent to the identity $I$, such that
$A_{n}=E_{n}AE_{n}$.

####  9.3 Continuous functional calculus

In the introduction we have noted that the matrix spectral theorem allows to
define the functional calculus of Hermitian matrices. Our strategy in the
Hilbert space operator case is opposite: we first built up a continuous
functional calculus (thanks to the the two above definition, and -of course-
to the matrix spectral theorem)

######  Lemma 9.3.1.

If $\\{A_{n}\\}_{n\in\mathbb{N}}$ is a basic sequence for
$A\in\mathbb{S}({\mathcal{H}})$ and $p(t)$ is a polynomial, then

${\mathrm{sot}}\lim_{n\to\infty}p(A_{n})=p(A).$

###### Proof.

∎

######  Lemma 9.3.2.

Let $f(t)\in C^{0}$ and let $A\in\mathbb{S}({\mathcal{H}})$. Then, there
exists an operator in $\mathbb{S}({\mathcal{H}})$, that we denote $f(A)$, such
that

$f(A)={\mathrm{sot}}\lim_{n\to\infty}f(A_{n})$ (9.3.1)

for any basic sequence $\\{A_{n}\\}_{n\in\mathbb{N}}$ for $A$. Moreover
$f(A)A=Af(A)$.

Thus, we have natural way to define operators $f(A)$ from continuous function
$f(t)$ and Hermitian arguments $A$. This is called the (operator) continous
functional calculus. Lemma 2.1 shows that for a polynomial, the continous
functional calculus yields, of course, the same operator as its algebraic
definition.

###### Proof.

Fix a unit vector $h\in{\mathcal{H}}$ and a scalar $\varepsilon$, and pick a
polynomial $p(t)$ such that $|f(t)-p(t)|<\varepsilon$ whenever $|t|\leq\|A\|$.
Since $\|A_{n}\|\leq\|A\|$, for any $n$, we infer

$\|f(A_{n})-p(A_{n})\|\leq\varepsilon.$ (9.3.2)

Let $n_{0}$ be an integer such that

$\|p(A_{n})h-p(A)h\|\leq\varepsilon$ (9.3.3)

for all $n\geq n_{0}$. Combining (9.3.2) and (9.3.3) we get
$\|f(A_{n})h-p(A)h\|\leq 2\varepsilon$ for all $n\geq n_{0}$. As $\varepsilon$
can be arbitrarily small, we obtain that $\\{f(A_{n})h\\}$ is a Cauchy
sequence in ${\mathcal{H}}$ and thus has a limit that we call $f(A)h$. It
obviously defines a linear operator as $f(A_{n})$ are linear operators. And
since $\|f(A_{n})\|\leq\max\\{|f(t)|:-\|A\|\leq t\leq\|A\|\\}$, $f(A)$ is a
bounded linear operator. Thus we have the strong limit in $\mathbb{S}$,
$f(A)={\mathrm{sot}}\lim_{n}f(A_{n}).$ To check that $f(A)$ does not depend on
the basic sequence of $A$, suppose that $\\{A^{\prime}_{n}\\}$ is another
basic sequence of $A$ and note that we also have
$\|f(A^{\prime}_{n})h-p(A)h\|\leq 2\varepsilon$ for $n$ large enough and thus
$\\{f^{\prime}(A_{n})\\}$ also converges to $f(A)h$. Finally, as
$f(A_{n})A_{n}=A_{n}f(A_{n})$, taking the strong limits, we infer
$f(A)A=Af(A)$. ∎

For a finite rank operator $Z\in\mathbb{S}({\mathcal{H}})$,
${\mathrm{rank}}\,Z=m$, with spectral decomposition

$Z=\sum_{j=1}^{m}z_{j}^{\uparrow}P_{j},$

where $z_{j}$ are the nonzero eigenvalues of $Z$ counted with their
multiplicities and $P_{j}$ are corresponding rank one spectral projections,
the functional calculus with $f\in C^{0}$ is obtained by the simple formulae

$f(Z)=f(0)P_{0}+\sum_{j=1}^{m}f(z_{j}^{\uparrow})P_{j},$ (9.3.4)

$P_{0}$ standing for the projection onto the nullspace of $Z$. Therefore,
given two continuous functions such that $f(t)\geq g(t)$, we have $f(Z)\geq
g(Z)$. Applying this to a basic sequence we obtain the following remark.

######  Remark 9.3.3.

If $f(t),g(t)\in C^{0}$ satysfy $f(t)\geq g(t)$, then $f(A)\geq g(A)$ for all
$A\in\mathbb{S}({\mathcal{H}})$.

The functional calculus for a finite rank operator (9.3.4) also shows that

$\langle h,f(Z)f\rangle\leq\sup_{t\in\mathbb{R}}f(t)$

for all unit vectors $h$. Applying this to a basic sequence provides the next
simple observation.

######  Remark 9.3.4.

If $f(t)\in C^{0}$ and $A\in\mathbb{S}({\mathcal{H}})$, then, for all unit
vectors $h$,

$\langle h,f(A)h\rangle\leq\sup_{t\in\mathbb{R}}f(t).$

####  9.4 Spectral projections

The aim of this section is to define the operator version of the matrix
spectral projections (9.2.2). We first give a useful fact, known as Vigier’s
theorem.

######  Lemma 9.4.1.

Let $\\{Z_{n}\\}_{n\in\mathbb{N}}$ be a decreasing sequence of positive
operators on ${\mathcal{H}}$, that is $Z_{n}\geq Z_{n+1}$ for all $n$. Then
there exists a positive operator $Z$ such that

$Z={\mathrm{sot}}\lim_{n\to\infty}Z_{n}$

###### Proof.

The wot convergence follows from polarization. Then, we may infer the sot
convergence. Indeed, given $h\in{\mathcal{H}}$, one has

$\displaystyle\|Z_{n+p}h-Z_{n}h\|^{2}$ $\displaystyle=\langle
h,(Z_{n}-Z_{n+p})^{2}h\rangle$ $\displaystyle=\langle
h,(Z_{n}-Z_{n+p})^{1/2}(Z_{n}-Z_{n+p})(Z_{n}-Z_{n+p})^{1/2}h\rangle$
$\displaystyle\leq\|Z_{n}-Z_{n+p}\|\langle h,(Z_{n}-Z_{n+p})h\rangle.$

∎

######  Lemma 9.4.2.

Let $A\in\mathbb{S}({\mathcal{H}})$. For $\lambda\in(-\infty,\infty)$ and
$n\in\mathbb{N}^{*}$, define the three piece affine continuous function
$f_{\lambda,n}(t)$ such that

$f_{\lambda,n}(t):=\begin{cases}1&\text{if \ $t\leq\lambda$},\\\
1+n(\lambda-t)&\text{if \ $\lambda\leq t\leq\lambda+n^{-1}$},\\\ 0&\text{if \
$\lambda+n^{-1}\leq t$}.\\\ \end{cases}$

Then, there exists a projection $E(\lambda)\in\mathbb{S}({\mathcal{H}})$
commuting with $A$ such that

$E(\lambda)={\mathrm{sot}}\lim_{n\to\infty}f_{\lambda,n}(A)$

and $\langle h,AE(\lambda)h\rangle\leq\lambda$ for all unit vectors $h$.

###### Proof.

Note that $f_{\lambda,n}(t)\geq f_{\lambda,n+1}(t)\geq 0$, $n\in\mathbb{N}$.
Remark 9.3.3 shows that $\\{f_{\lambda,n}(A)\\}_{n\in\mathbb{N}}$ is a
decreasing sequence of positive operators. Thanks to Lemma 9.3.2 we thus have
a positive operator $E(\lambda)$ as the strong limit

$E(\lambda)={\mathrm{sot}}\lim_{n\to\infty}f_{\lambda,n}(A).$

Since $f_{\lambda,n}(A)A=Af_{\lambda,n}(A)$, we also have the commutativity of
$A$ with this strong limit, $E(\lambda)A=AE(\lambda)$. Further, by Remark
9.3.4, given a unit vectors $h$,

$\langle
h,Af_{\lambda,n}(A)h\rangle\leq\sup_{t\in\mathbb{R}}tf_{\lambda,n}(t)$

As the supremum tends to $\lambda$ as $n\to\infty$, we get $\langle
h,AE(\lambda)h\rangle\leq\lambda$ for all unit vectors $h$.

Since $1\geq f_{\lambda,n}(t)$, Remark 9.3.3 shows that $I\geq E(\lambda)$ and
hence $E(\lambda)\geq E^{2}(\lambda)$. We also have $E^{2}(\lambda)\geq
E(\lambda)$, indeed,

$\displaystyle E^{2}(\lambda)$
$\displaystyle=\left({\mathrm{sot}}\lim_{n\to\infty}f_{\lambda,n}(A)\right)^{2}$
$\displaystyle={\mathrm{sot}}\lim_{n\to\infty}f^{2}_{\lambda,n}(A)$
$\displaystyle\geq{\mathrm{sot}}\lim_{m\to\infty}f_{\lambda,m}(A)=E(\lambda)$

as for each integer $n$ there exists an integer $m$ such that
$f^{2}_{\lambda,n}(t)\geq f_{\lambda,m}(t)$. Thus $E^{2}(\lambda)=E(\lambda)$,
i.e., $E(\lambda)$ is a projection. ∎

######  Remark 9.4.3.

The projection valued map $\lambda\to E(\lambda)$ is increasing and sot right
continuous, we call this map (as well as the family $\\{E(\lambda)\\}$ with
$\lambda\in\mathbb{R})$ the spectral measure of $A$. The previous lemma shows
that given two self-adjoint operators $A$ and $B$ with respectives spectral
measures $\\{E(\lambda)\\}$ and $\\{F(\nu)\\}$, then the commutativity
assumption $AB=BA$ ensures the commutativity of the spectral measures:
$E(\lambda)F(\nu)=F(\nu)E(\lambda)$.

####  9.5 The spectral theorem

In case of $A\in\mathbb{S}({\mathcal{H}})$, the matrix decomposition (9.2.3)
(involving the matrix spectral measure….) remains valid in a continuous form:

######  Theorem 9.5.1.

Let $A\in\mathbb{S}({\mathcal{H}})$ with its spectral measure $E(\lambda)$.
Then

$A=\int\lambda\,{\mathrm{d}}E(\lambda)$

Now we explain the Stieltjes-integral notation employed, and next give the
proof which is a straightforward consequence of the existence of the spectral
measure built up in the previous Lemma.

We may detail a little bit more the theorem, The spectral measure vanishes on
an open set $\omega(A)$ of $\mathbb{R}$:

$x\in\omega(A)\iff\exists\varepsilon>0\ {\mathrm{s.t.}}\
E(x+\varepsilon)-E(x-\varepsilon)=0.$

The complementary set of $\omega$ is the compact set $\sigma(A)$, the spectrum
of $A$. Then we may write the spectral theorem as

######  Theorem 9.5.2.

Let $A\in\mathbb{S}({\mathcal{H}})$ with its spectral measure $E(\lambda)$.
Then

$A=\int_{\sigma(A)}\lambda\,{\mathrm{d}}E(\lambda).$

Denote by $C^{*}(A)$ the unital $C^{*}$-algebra spanned by $A$. We have the
following concrete version of the Gelfand isomorphism mentionned in the
introduction.

######  Corollary 9.5.3.

Let $A\in\mathbb{S}({\mathcal{H}})$. Then the functional calculus from
$C^{0}(\sigma(A))$ to $C^{*}(A)$,

$f(t)\mapsto f(A):=\int_{\sigma(A)}f(\lambda)\,{\mathrm{d}}E(\lambda).$

is an isometric $*$-isomorphism.

From Theorem 9.5.1 and Remark 9.4.3 we get:

######  Corollary 9.5.4.

Let $A,B\in\mathbb{S}({\mathcal{H}})$ be commuting. Then
$AB\in\mathbb{S}({\mathcal{H}})$.

Taking advantage that a normal operator $N$ has a decomposition $N=A+iB$ in
which $A$ and $B$ are two commuting self-adjoint operators, the above
corollary and Remark 9.4.3 show that $A$ and $B$ have two commuting respective
spectral measures, say $E(x)$ and $F(y)$, $x,y\in\mathbb{R}$. Thus we obtain a
spectral measure $G(z)$, $z\in\mathbb{C}$, for $N$ and get the following
spectral theorem for normal operators:

######  Theorem 9.5.5.

Let $N$ be a normal operator with its spectral measure $G(\lambda)$. Then

$N=\int_{\mathbb{C}}\lambda\,{\mathrm{d}}G(\lambda)=\int_{\sigma(N)}\lambda\,{\mathrm{d}}G(\lambda)$

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