$\begin{bmatrix}A&0\\\ 0&B\end{bmatrix}=U_{1}\begin{bmatrix}(1-x)A+xB&0\\\
0&0\end{bmatrix}U_{1}^{*}+V_{1}\begin{bmatrix}0&0\\\
0&xA+(1-x)B\end{bmatrix}V_{1}^{*}$ (4.2.2)

for two unitary matrices $U_{1},V_{1}\in\mathbb{M}_{2n}$.

Next, recall the subadditivity inequality [31, Theorem 3.4] : Given any pair
of positive semidefinite matrices $S,T\in\mathbb{M}_{d}$, we have

$f(S+T)\leq U_{2}f(S)U_{2}^{*}+V_{2}f(T)V_{2}^{*}$

for two unitary matrices $U_{2},V_{2}\in\mathbb{M}_{d}$. Applying this to
(4.2.2) yields

$\begin{bmatrix}f(A)&0\\\ 0&f(B)\end{bmatrix}\leq
U\begin{bmatrix}f((1-x)A+xB)&0\\\
0&f(0)I\end{bmatrix}U^{*}+V\begin{bmatrix}f(0)I&0\\\
0&f(xA+(1-x)B)\end{bmatrix}V^{*}$

for two unitary matrices $U,V\in\mathbb{M}_{2n}$. This proves the theorem when
$f(0)=0$.

To derive the general case, we may assume that $f(t)$ is continuous (a concave
function on $[0,\infty)$ might be discontinuous at $0$). Indeed, it suffices
to consider the values of $f(t)$ on a finite set, the union of the spectra of
the four matrices $A$, $B$, $(1-x)A+xB$ and $xA+(1-x)B$. Hence we may replace
$f(t)$ by a piecewise affine monotone concave function $h(t)$ with $h(0)\geq
0$. Now, since $h(t)$ is continuous, a limit argument allows us to suppose
that $A$ and $B$ are invertible. Therefore, letting
$\lambda_{n}^{\downarrow}(Z)$ denote the smallest eigenvalue of
$Z\in\mathbb{M}_{n}^{s.a}$,

$r:=\min\\{\lambda_{n}^{\downarrow}(A),\lambda_{n}^{\downarrow}(B)\\}>0.$

We may then replace $h(t)$ by $h_{r}(t)$ defined as $h_{r}(t):=h(t)$ for
$t\geq r$, $h_{r}(0):=0$ and $h_{r}(s):=h(r)\frac{s}{r}$ for $0\leq s\leq r$.
The function $h_{r}(t)$ is monotone concave on $[0,\infty)$ and vanishes at
$0$, thus the case $f(0)=0$ entails the general case. ∎

Let $\lambda_{j}^{\downarrow}(Z)$, $j=1,2,\ldots n$, denote the eigenvalues of
$Z\in\mathbb{M}_{n}^{s.a}$ arranged in the nonincreasing order.

######  Corollary 4.2.2.

Let $A,B\in\mathbb{M}_{n}^{+}$, let $0<x<1$, and let $f(t)$ be a nonnegative
concave function on $[0,\infty)$. Then, for $j=0,1,\ldots,n-1$,

$\lambda_{1+2j}^{\downarrow}\left(f(A\oplus
B)\right)\leq\lambda_{1+j}^{\downarrow}\left(f(xA+(1-x)B)\right)+\lambda_{1+j}^{\downarrow}\left(f((1-x)A+xB)\right)$

and

$\lambda_{1+j}^{\downarrow}\left\\{\left(f(xA+(1-x)B)\right)+\left(f((1-x)A+xB)\right)\right\\}\leq
2\lambda_{1+j}^{\downarrow}\left(f\left(\frac{A+B}{2}\right)\right).$

###### Proof.

The first inequality is a straighforward consequence of Theorem 4.2.1 combined
with the inequalities of Weyl [13, p. 62] : For all
$S,T\in\mathbb{M}_{d}^{s.a}$ and $j,k\in\\{0,\ldots,d-1\\}$ such that
$j+k+1\leq d$,

$\lambda_{1+j+k}^{\downarrow}(S+T)\leq\lambda_{1+j}^{\downarrow}(S)+\lambda_{1+k}^{\downarrow}(T).$

The second inequality is not new; it follows from Theorem 8.5.1. ∎

######  Corollary 4.2.3.

Let $A,B\in\mathbb{M}_{n}^{+}$, let $0<x<1$, and let $f(t)$ be nonnegative
concave function on $[0,\infty)$. Then, for all $p\geq 1$,

$\left(\|f(A)\|_{p}^{p}+\|f(B)\|_{p}^{p}\right)^{1/p}\leq\|f(xA+(1-x)B))\|_{p}+\|f((1-x)A+xB)\|_{p}.$

###### Proof.

From Theorem 4.2.1 we have

$\|f(A)\oplus f(B)\|_{p}\leq\|Uf((1-x)A+xB)U^{*}+Vf(xA+(1-x)B)V^{*}\|_{p}.$

The triangle inequality for $\|\cdot\|_{p}$ completes the proof. ∎

Corollary 4.2.3 with $f(t)=t^{q}$ reads as the following trace inequality.

######  Corollary 4.2.4.

Let $A,B\in\mathbb{M}_{n}^{+}$ and $0<x<1$. Then, for all $p\geq 1\geq q\geq
0$,

$\left\\{{\mathrm{Tr\,}}A^{pq}+\mathrm{Tr\,}B^{pq}\right\\}^{1/p}\leq\left\\{{\mathrm{Tr\,}}(xA+(1-x)B)^{pq}\right\\}^{1/p}+\left\\{{\mathrm{Tr\,}}((1-x)A+xB))^{pq}\right\\}^{1/p}.$

Choosing in Corollary 4.2.4 $q=1$ and $x=1/2$ yields McCarthy’s inequality,

${\mathrm{Tr\,}}A^{p}+{\mathrm{Tr\,}}B^{p}\leq{\mathrm{Tr\,}}(A+B)^{p}.$

This shows that Theorem 4.2.1 is already significant with $f(t)=t$. Our next
corollary, for convex functions, is equivalent to Theorem 4.2.1.

######  Corollary 4.2.5.

Let $A,B\in\mathbb{M}_{n}^{+}$, let $0<x<1$, and let $g(t)$ be a monotone
convex function on $[0,\infty)$ with $g(0)\leq 0$. Then, for some isometry
matrices $U,V\in\mathbb{M}_{2n,n}$,

$g(A)\oplus g(B)\geq Ug((1-x)A+xB)U^{*}+Vg(xA+(1-x)B)V^{*}.$

Since the Schatten $q$-quasinorms $\|\cdot\|_{q}$, $0<q<1$, are superadditive
functionals on $\mathbb{M}_{n}^{+}$, Corollary 4.2.5 yields the next one.

######  Corollary 4.2.6.

Let $A,B\in\mathbb{M}_{n}^{+}$, let $0<x<1$, and let $g(t)$ be a nonnegative
convex function on $[0,\infty)$ with $g(0)\leq 0$. Then, for all $0<q<1$,

$\left(\|g(A)\|_{q}^{q}+\|g(B)\|_{q}^{q}\right)^{1/q}\geq\|g(xA+(1-x)B))\|_{q}+\|g((1-x)A+xB)\|_{q}.$

From the first inequality of Corollary 4.2.2 we also get the following
statement.

######  Corollary 4.2.7.

Let $A,B\in\mathbb{M}_{n}^{+}$ and let $f(t)$ be a nonnegative concave
function on $[0,\infty)$. Then, for $j=0,1,\ldots,n-1$,

$\lambda_{1+2j}^{\downarrow}\left(f(A\oplus B)\right)\leq
2\int_{0}^{1}\lambda_{1+j}^{\downarrow}\left(f(xA+(1-x)B)\right)\,{\mathrm{d}}x.$

Up to now we have dealt with convex combinations $(1-x)A+xB$ with scalar
weights. It is natural to search for extensions with matricial weights
($C^{*}$-convex combinations). We may generalize Theorem 4.2.1 with commuting
normal weights.

######  Theorem 4.2.8.

Let $A,B\in\mathbb{M}_{n}^{+}$ and let $f(t)$ be a monotone concave function
on $[0,\infty)$ with $f(0)\geq 0$. If $X,Y\in\mathbb{M}_{n}$ are normal and
satisfy $XY=YX$ and $X^{*}X+Y^{*}Y=I$, then, for some isometry matrices
$U,V\in\mathbb{M}_{2n,n}$,

$f(A)\oplus f(B)\leq Uf(X^{*}AX+Y^{*}BY)U^{*}+Vf(Y^{*}AY+X^{*}BX)V^{*}.$

###### Proof.

The proof is quite similar to that of Theorem 4.2.1 except that we first
observe that the $2n\times 2n$ matrix

$H:=\begin{bmatrix}X&Y\\\ Y&-X\end{bmatrix}$

is unitary. Indeed for two normal operators, $XY=YX$ ensures $X^{*}Y=YX^{*}$
and a direct computation shows that $H^{*}H$ is the identity in
$\mathbb{M}_{2n}$. We then use the unitary congruence

$H^{*}\begin{bmatrix}A&0\\\
0&B\end{bmatrix}H=\begin{bmatrix}X^{*}AX+Y^{*}BY&\star\\\
\star&Y^{*}AY+X^{*}BX\end{bmatrix}$ (4.2.3)

where the stars hold for unspecified entries. ∎

Hence, in the first inequality of Corolloray 4.2.2 and in the series of
Corollaries 4.2.3-4.2.6, we can replace the scalar convex combinations
$(1-x)A+B$ and $xA+(1-x)B$ by $C^{*}$-convex combinations $X^{*}AX+Y^{*}BY$
and $Y^{*}AY+X^{*}BX$ with commuting normal weights. Here we explicitly state
the generalization of Corollary 4.2.5.

######  Corollary 4.2.9.

Let $A,B\in\mathbb{M}_{n}^{+}$ and let $g(t)$ be a monotone convex function on
$[0,\infty)$ with $g(0)\leq 0$. If $X,Y\in\mathbb{M}_{n}$ are normal and
satisfy $XY=YX$ and $X^{*}X+Y^{*}Y=I$, then, for some isometry matrices
$U,V\in\mathbb{M}_{2n,n}$,

$g(A)\oplus g(B)\geq Ug(X^{*}AX+Y^{*}BY)U^{*}+Vg(Y^{*}AY+X^{*}BX)V^{*}.$

####  4.3 Majorization

The results of Section 2 require two essential assumptions : to deal with
positive matrices and with subadditive (concave) or superadditive (convex)
functions. Thanks to these assumptions, we have obtained operator inequalities
for the usual order in the positive cone.

The results of this section will consider Hermitian matrices and general
convex or concave functions. We will obtain majorization relations. We also
consider $C^{*}$-convex combinations more general than those with commuting
normal weights.

We recall the notion of majorization. Let $A,B\in\mathbb{M}_{n}^{s.a}$. We say
that $A$ is weakly majorized by $B$ and we write $A\prec_{w}B$, if

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

for all $k=1,2,\ldots n$. If furthemore the equality holds for $k=n$, that is
$A$ and $B$ have the same trace, then we say that $A$ is majorized by $B$,
written $A\prec B$. See [13, Chapter 2] and [64] for a background on
majorization. One easily checks that $A\prec_{w}B$ is equivalent to $A+C\prec
B$ for some $C\in\mathbb{M}_{n}^{+}$. We need two fundamental principles:

  * (1)

$A\prec B\Rightarrow g(A)\prec_{w}g(B)$ for all convex functions $g(t)$.

  * (2)

$A\prec_{w}B\iff{\mathrm{Tr\,}}f(A)\leq{\mathrm{Tr\,}}f(B)$ for all
nondecreasing convex functions $f(t)$. Equivalently $A\prec
B\iff{\mathrm{Tr\,}}g(A)\leq{\mathrm{Tr\,}}g(B)$ for all convex functions
$g(t)$.

######  Lemma 4.3.1.

Let $A,B\in\mathbb{M}_{n}^{s.a}$ and let $g(t)$ be a convex function defined
on an interval containing the spectra of $A$ and $B$. If
$X,Y\in\mathbb{M}_{n}$ satisfy $X^{*}X+Y^{*}Y=XX^{*}+YY^{*}=I$, then,

${\mathrm{Tr\,}}\left\\{g(X^{*}AX+Y^{*}BY)+g(Y^{*}AY+X^{*}BX)\right\\}\leq{\mathrm{Tr\,}}\left\\{g(A)+g(B)\right\\}.$

###### Proof.

By the famous Hansen-Pedersen trace inequality [67], see also [31, Corollary
2.4] for a generalization,

$\displaystyle{\mathrm{Tr\,}}$
$\displaystyle\left\\{g(X^{*}AX+Y^{*}BY)+g(Y^{*}AY+X^{*}BX)\right\\}$
$\displaystyle\leq{\mathrm{Tr\,}}\left\\{X^{*}g(A)X+Y^{*}g(B)Y)+Y^{*}g(A)Y+X^{*}g(B)X\right\\}$
$\displaystyle={\mathrm{Tr\,}}\left\\{(g(A)+g(B))(XX^{*}+YY^{*})\right\\}={\mathrm{Tr\,}}\left\\{g(A)+g(B)\right\\}$

where the first equality follows from the cyclicity of the trace. ∎

######  Theorem 4.3.2.

Let $A,B\in\mathbb{M}_{n}^{s.a}$ and $X,Y\in\mathbb{M}_{n}$. If
$X^{*}X+Y^{*}Y=XX^{*}+YY^{*}=I$, then, for some unitary matrices
$\\{U_{k}\\}_{k=1}^{2n}$ in $\mathbb{M}_{2n}$,

$(X^{*}AX+Y^{*}BY)\oplus(Y^{*}AY+X^{*}BX)=\frac{1}{2n}\sum_{k=1}^{2n}U_{k}(A\oplus
B)U_{k}^{*}.$

###### Proof.

From Lemma 4.3.1, we have the trace inequality

${\mathrm{Tr\,}}g\left((X^{*}AX+Y^{*}BY)\oplus(Y^{*}AY+X^{*}BX)\right)\leq{\mathrm{Tr\,}}g(A\oplus
B)$

for all convex functions defined on $(-\infty,\infty)$. By a basic principle
of majorization, this is equivalent to

$(X^{*}AX+Y^{*}BY)\oplus(Y^{*}AY+X^{*}BX)\prec A\oplus B.$ (4.3.1)

By [42, Proposition 2.6], the majorization in $\mathbb{M}_{d}^{s.a}$, $S\prec
T$, ensures that (and thus is equivalent to)

$S\leq\frac{1}{d}\sum_{j=1}^{d}V_{j}TV_{j}^{*}$

for $d$ unitary matrices $V_{j}\in\mathbb{M}_{d}$. Applying this to (4.3.1)
completes the proof. ∎

######  Remark 4.3.3.

We can prove the majorization (4.3.1) in a different way by oberving that our
assumption on $X$ and $Y$ ensures that the map $\Phi$, defined on
$\mathbb{M}_{d}$, (here $d=2n$),

$\Phi\left(\begin{bmatrix}A&C\\\
D&B\end{bmatrix}\right):=\begin{bmatrix}X^{*}AX+Y^{*}BY&0\\\
0&Y^{*}AY+X^{*}BX)\end{bmatrix},$

is a positive linear map unital and trace preserving. Such maps are also
called doubly stochastic. It is a classical result (see Ando’s survey [5,
Section 7] and references therein) that we have

$\Phi(Z)\prec Z$ (4.3.2)

for every doubly stochastic map on $\mathbb{M}_{d}$ and Hermitian $Z$. Here,
our map is even completely positive (it is a so called quantum channel). It is
well known in the litterature ([5, Theorem 7.1]) that we have then

$\Phi(Z)=\sum_{j=1}^{m}t_{j}U_{j}^{*}ZU_{j}$

for some convex combination $0<t_{j}\leq 1$, $\sum_{j=1}^{m}t_{j}=1$, and some
unitary matrices $U_{j}$ (these scalars $t_{i}$ and matrices $U_{i}$ depend on
$Z$). In 2003, Zhan [94] noted that we can take $m=d$. That we can actually
take an average,

$\Phi(Z)=\frac{1}{d}\sum_{j=1}^{d}U_{j}^{*}ZU_{j},$

follows from the quite recent observation [42, Proposition 2.6] mentioned in
the proof of Theorem 4.3.2. For quantum channels, one has the Choi-Kraus
decomposition (see [15, Chapter 3])

$\Phi(A)=\sum_{i=1}^{d^{2}}K_{i}AK_{i}^{*}$

for some weights $K_{i}\in\mathbb{M}_{d}$ such that
$\sum_{i=1}^{d^{2}}K_{i}K_{i}^{*}=\sum_{i=1}^{d^{2}}K_{i}^{*}K_{i}=I$. So, the
proof of Lemma 3.1 shows that, for quantum channels, one may derive the
fundamental majorization (4.3.2) from two results for convex functions: the
basic principle of majorisation and Hansen-Pedersen’s trace inequality.

Theorem 4.3.2 combined with Theorem 4.1.2 provide a number of interesting
operator inequalities. The next corollary can be regarded as another matrix
version of the scalar inequality (4.1.2). We will give a proof independent of
Theorem 4.1.2.

######  Corollary 4.3.4.

Let $A,B\in\mathbb{M}_{n}^{s.a}$ and let $f(t)$ be a convex function defined
on an interval containing the spectra of $A$ and $B$. If
$X,Y\in\mathbb{M}_{n}$ satisfy $X^{*}X+Y^{*}Y=XX^{*}+YY^{*}=I$, then, for some
unitary matrices $\\{U_{k}\\}_{k=1}^{2n}$ in $\mathbb{M}_{2n}$,

$f\left(X^{*}AX+Y^{*}BY\right)\oplus
f\left(Y^{*}AY+X^{*}BX\right)\leq\frac{1}{2n}\sum_{k=1}^{2n}U_{k}f\left(A\oplus
B\right)U_{k}^{*}.$

In particular, for the absolute value, we note that

$\left|X^{*}AX+Y^{*}BY\right|\oplus\left|Y^{*}AY+X^{*}BX\right|\leq\frac{1}{2n}\sum_{k=1}^{4n}U_{k}\left|A\oplus
B\right|U_{k}^{*}.$

###### Proof.

The majorization (4.3.1) and the basic principle of majorizations show that

$f\left(X^{*}AX+Y^{*}BY\right)\oplus
f\left(Y^{*}AY+X^{*}BX\right)\prec_{w}f(A\oplus B)$

for all convex functions defined on an interval containing the spectra of $A$
and $B$. Thus

$f\left(X^{*}AX+Y^{*}BY\right)\oplus f\left(Y^{*}AY+X^{*}BX\right)+C\prec
f(A\oplus B)$

for some positive semidefinte matrice $C\in\mathbb{M}_{2n}^{+}$. Hence, as in
the previous proof,

$f\left(X^{*}AX+Y^{*}BY\right)\oplus
f\left(Y^{*}AY+X^{*}BX\right)+C\leq\frac{1}{2n}\sum_{k=1}^{2n}U_{k}f\left(A\oplus
B\right)U_{k}^{*}$

for some family of unitary matrices $U_{k}\in\mathbb{M}_{2n}$. ∎

######  Corollary 4.3.5.

Let $A,B\in\mathbb{M}_{n}^{+}$. If $X,Y\in\mathbb{M}_{n}$ satisfy
$X^{*}X+Y^{*}Y=XX^{*}+YY^{*}=I$, then,

$\det(X^{*}AX+Y^{*}BY)\det(Y^{*}AY+X^{*}BX)\geq\det A\det B.$

###### Proof.

Since the classical Minkowski functional $Z\mapsto\det^{1/2n}Z$ is concave on
$\mathbb{M}_{2n}^{+}$, the result is an immediate consequence of Theorem
4.3.2. ∎

The case $A=B$ and $X,Y$ are two orthogonal projections reads as the classical
Fisher’s inequality.

Corollary 4.3.4 yields inequalities for symmetric norms and antinorms on
$\mathbb{M}_{2n}^{+}$. Symmetric norms, $\|\cdot\|$, also called unitarily
invariant norms, are classical objects in matrix analysis. We refer to [13],
[64], [67, Chapter 6] and, in the setting of compact operators, [89]. The most
famous examples are the Schatten $p$-norms, $1\leq p\leq\infty$, and the Ky
Fan $k$-norms.

Symmetric anti-norms $\|\cdot\|_{!}$ are the concave counterpart of symmetric
norms. Famous examples are the Schatten $q$-quasi norms, $0<q<1$ and the
Minkowski functional considered in the proof of Corollary 4.3.5. We refer to
[28] and [29, Section 4] for much more examples.

######  Corollary 4.3.6.

Let $f(t)$ and $g(t)$ be two nonnegative functions defined on $[a,b]$ and let
$A,B\in\mathbb{M}_{n}$ be Hermitian with spectra in $[a,b]$. If
$X,Y\in\mathbb{M}_{n}$ satisfy $X^{*}X+Y^{*}Y=XX^{*}+YY^{*}=I$, then :

  * (i)

If $f(t)$ is concave, then, for all symmetric antinorms,

$\left\|f\left(X^{*}AX+Y^{*}BY\right)\oplus
f\left(Y^{*}AY+X^{*}BX\right)\right\|_{!}\geq\left\|f\left(A\oplus
B\right)\right\|_{!}.$

  * (ii)

If $g(t)$ is convex, then, for all symmetric norms,

$\left\|g\left(X^{*}AX+Y^{*}BY\right)\oplus
g\left(Y^{*}AY+X^{*}BX\right)\right\|\leq\left\|g\left(A\oplus
B\right)\right\|.$

###### Proof.

Since symmetric antinorms are unitarily invariant and superadditive, the first
assertion follows from the version of Corollary 4.3.4 for concave version. The
second assertion is an immediate consequence of Corollary 4.3.4. ∎

We close this section by mentioning Moslehian’s weak majorization which
provides a matrix version of the first inequality of Proposition 4.1.1. We may
restate [81, Corollary 3.4] as inequalities for symmetric and antisymmetric
norms.

######  Proposition 4.3.7.

Let $f(t)$ and $g(t)$ be two nonnegative functions defined on $[a,b]$ and let
$A,B\in\mathbb{M}_{n}$ be Hermitian with spectra in $[a,b]$.

  * (i)

If $f(t)$ is concave, then, for all symmetric antinorms,

$\left\|f\left(\frac{A+B}{2}\right)\right\|_{!}\geq\left\|\int_{0}^{1}f((1-x)A+xB)\,{\mathrm{d}}x\right\|_{!}.$

  * (ii)

If $g(t)$ is convex, then, for all symmetric norms,

$\left\|g\left(\frac{A+B}{2}\right)\right\|\leq\left\|\int_{0}^{1}g((1-x)A+xB)\,{\mathrm{d}}x\right\|.$

####  4.4 References of chapter 4

  * [5

] T. Ando, Majorization, doubly stochastic matrices, and comparison of
eigenvalues, Linear Algebra Appl.  118 (1989), 163-248.

  * [13

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

  * [15

] R. Bhatia, Positive Definite Matrices, Princeton University press, Princeton
2007.

  * [28

] J.-C. Bourin and F. Hiai, Norm and anti-norm inequalities for positive semi-
definite matrices, Internat. J. Math. 63 (2011), 1121-1138.

  * [29

] J.-C. Bourin and F. Hiai, Jensen and Minkowski inequalities for operator
means and anti-norms. Linear Algebra Appl. 456 (2014), 22–53.

  * [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.

  * [42

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

  * [43

] J.-C. Bourin and E.-Y. Lee, Matrix inequalities and majorizations around
Hermite-Hadamard’s inequality, preprint

  * [64

] F. Hiai, Matrix Analysis: Matrix Monotone Functions, Matrix Means, and
Majorization (GSIS selected lectures), Interdisciplinary Information Sciences
16 (2010), 139–248.

  * [67

] F. Hiai, D. Petz, Introduction to Matrix Analysis and applications.
Universitext, Springer, New Delhi, 2014.

  * [81

] M. S. Moslehian, Matrix Hermite-Hadamard type inequalities, Houston J. Math.
39 (2013), no. 1, 177–189.

  * [83

] C. P. Niculescu; L.-E. Persson, Old and new on the Hermite-Hadamard
inequality, Real Anal. Exchange 29 (2003/04), no. 2, 663–685.

  * [89

] B. Simon, Trace ideal and their applications, Cambridge University Press,
Cambridge, 1979.

  * [94

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

##  Part II Operator diagonals and partitionned matrices

$\begin{bmatrix}A&X\\\ X&B\end{bmatrix}=\frac{U(A+B)U^{*}+V(A+B)V^{*}}{2}$

###  Chapter 5 Pinchings and Masas

For a background on the essential numerical and the proof of the pinching
theorem used in this article, see Section 5.6 “Around this article”.

Pinchings and positive linear maps [37]

Abstract. We employ the pinching theorem, ensuring that some operators $A$
admit any sequence of contractions as an operator diagonal of $A$, to
deduce/improve two recent theorems of Kennedy-Skoufranis and Loreaux-Weiss for
conditional expectations onto a masa in the algebra of operators on a Hilbert
space. Similarly, we obtain a proof of a theorem of Akeman and Anderson
showing that positive contractions in a continuous masa can be lifted to a
projection. We also discuss a few corollaries for sums of two operators in the
same unitary orbit.

Keywords: Pinching, essential numerical range, positive linear maps,
conditional expectation onto a masa, unitary orbit. 2010 Mathematics Subject
Classification. 46L10, 47A20, 47A12.

####  5.1 The pinching theorem

We recall two theorems which are fundamental in the next sections to obtain
several results about positive linear maps, in particular conditional
expectations, and unitary orbits. These theorems were established in [21], we
also refer to this article for various definitions and properties of the
essential numerical range $W_{e}(A)$ of an operator $A$ in the algebra
${\mathrm{L}}({\mathcal{H}})$ of all (bounded linear) operators on an infinite
dimensional, separable (real or complex) Hilbert space ${\mathcal{H}}$.

We denote by ${\mathcal{D}}$ the unit disc of $\mathbb{C}$. We write $A\simeq
B$ to mean that the operators $A$ and $B$ are unitarily equivalent. This
relation is extended to operators possibly acting on different Hilbert spaces,
typically, $A$ acts on ${\mathcal{H}}$ and $B$ acts on an infinite dimensional
subspace ${\mathcal{S}}$ of ${\mathcal{H}}$, or on the spaces
${\mathcal{H}}\oplus{\mathcal{H}}$ or $\oplus^{\infty}{\mathcal{H}}$.

######  Theorem 5.1.1.

Let $A\in{\mathrm{L}}({\mathcal{H}})$ with $W_{e}(A)\supset{\mathcal{D}}$ and
$\\{X_{i}\\}_{i=1}^{\infty}$ a sequence in ${\mathrm{L}}({\mathcal{H}})$ such
that $\sup_{i}\|X_{i}\|<1$. Then, a decomposition
${\mathcal{H}}=\bigoplus_{i=1}^{\infty}{\mathcal{H}}_{i}$ holds with
$A_{{\mathcal{H}}_{i}}\simeq X_{i}$ for all $i$.

Of course, the direct sum refers to an orthogonal decomposition, and
$A_{{\mathcal{H}}_{i}}$ stands for the compression of $A$ onto the subspace
${\mathcal{H}}_{i}$.

Theorem 5.1.1 tells us that we have a unitary congruence between an operator
in ${\mathrm{L}}(\oplus^{\infty}{\mathcal{H}})$ and a “pinching” of $A$,

$\bigoplus_{i=1}^{\infty}X_{i}\simeq\sum_{i=1}^{\infty}E_{i}AE_{i}$

for some sequence of mutually orthogonal infinite dimensional projections
$\\{E_{i}\\}_{i=1}^{\infty}$ in ${\mathrm{L}}({\mathcal{H}})$ summing up to
the identity $I$. Thus $\\{X_{i}\\}_{i=1}^{\infty}$ can be regarded as an
operator diagonal of $A$. In particular, if $X$ is an operator on
${\mathcal{H}}$ with $\|X\|<1$, then, $A$ is unitarily congruent to an
operator on ${\mathcal{H}}\oplus{\mathcal{H}}$ of the form,

$A\simeq\begin{pmatrix}X&\ast\\\ \ast&\ast\end{pmatrix}.$ (5.1.1)

For a sequence of normal operators, Theorem 5.1.1 admits a variation. Given
${\mathcal{A}},{\mathcal{B}}\subset\mathbb{C}$, the notation
${\mathcal{A}}\subset_{st}{\mathcal{B}}$ means that
${\mathcal{A}}+r{\mathcal{D}}\subset{\mathcal{B}}$ for some $r>0$.

######  Theorem 5.1.2.

Let $A\in{\mathrm{L}}({\mathcal{H}})$ and let $\\{X_{i}\\}_{i=1}^{\infty}$ be
a sequence of normal operators in ${\mathrm{L}}({\mathcal{H}})$ such that
$\cup_{i=1}^{\infty}W(X_{i})\subset_{st}W_{e}(A)$. Then, a decomposition
${\mathcal{H}}=\bigoplus_{i=1}^{\infty}{\mathcal{H}}_{i}$ holds with
$A_{{\mathcal{H}}_{i}}\simeq X_{i}$ for all $i$.

If all the operators are self-adjoint, this is true for the strict inclusion
in $\mathbb{R}$ (if ${\mathcal{A}},{\mathcal{B}}\subset\mathbb{R}$, the
notation ${\mathcal{A}}\subset_{st}{\mathcal{B}}$ then means that
${\mathcal{A}}+r{\mathcal{I}}\subset{\mathcal{B}}$ for some $r>0$, where
${\mathcal{I}}=[-1,1]$). This is actually an easy consequence of Theorem 5.1.1
or Theorem 5.1.2.

######  Corollary 5.1.3.

Let $A\in{\mathrm{L}}({\mathcal{H}})$ be self-adjoint and let
$\\{X_{i}\\}_{i=1}^{\infty}$ be a sequence of self-adjoint operators in
${\mathrm{L}}({\mathcal{H}})$ such that
$\cup_{i=1}^{\infty}W(X_{i})\subset_{st}W_{e}(A)$. Then, a decomposition
${\mathcal{H}}=\bigoplus_{i=1}^{\infty}{\mathcal{H}}_{i}$ holds with
$A_{{\mathcal{H}}_{i}}\simeq X_{i}$ for all $i$.

To get Corollary 5.1.3 from Theorem 5.1.2, let $W_{e}(A)=[a,b]$ and write
$A=A_{1}+A_{2}+A_{3}+A_{4}$ where $A_{i}A_{j}=0$ if $i\neq j$, with
$a\in\sigma_{e}(A_{1})\cap\sigma_{e}(A_{2})$ and
$b\in\sigma_{e}(A_{3})\cap\sigma_{e}(A_{4})$. Apply Theorem 5.1.2 to the
normal operator $\tilde{A}=(1+i)A_{1}+(1-i)A_{2}+(1+i)A_{3}+(1-i)A_{4}$ as
$\cup_{i=1}^{\infty}W(X_{i})\subset_{st}{\mathrm{conv}}\\{(1\pm i)a;(1\pm
i)b\\}\subset W_{e}(\tilde{A})$. We get a decomposition
${\mathcal{H}}=\bigoplus_{i=1}^{\infty}{\mathcal{H}}_{i}$ with
$\tilde{A}_{{\mathcal{H}}_{i}}\simeq X_{i}$ for all $i$. Therefore, taking
real parts we also have $A_{{\mathcal{H}}_{i}}\simeq X_{i}$.

In Section 3, our concern is the study of generalized diagonals, i.e.,
conditional expectations onto a masa in ${\mathrm{L}}({\mathcal{H}})$, of the
unitary orbit of an operator. The pinching theorems are the good tools for
this study; we easily obtain and considerably improve two recent theorems, of
Kennedy and Skoufranis for normal operators, and Loreaux and Weiss for
idempotent operators. For self-adjoint idempotents, i.e., projections, and
continuous masas, we obtain a theorem due to Akemann and Anderson. Section 4
deals with an application to the class of unital, positive linear maps which
are trace preserving. Section 5 collects a few questions on possible extension
of Theorems 5.1.1 and 5.1.2 in the setting of von Neumann algebras.

The next section gives applications which only require (5.1.1). These results
mainly focus on sums of two operators in a unitary orbit.

####  5.2 Sums in a unitary orbit

We recall a straightforward consequence of (5.1.1) for the weak convergence,
[21, Corollary 2.4].

######  Corollary 5.2.1.

Let $A,X\in{\mathrm{L}}({\mathcal{H}})$ with $W_{e}(A)\supset{\mathcal{D}}$
and $\|X\|\leq 1$. Then there exists a sequence of unitaries
$\\{U_{n}\\}_{n=1}^{\infty}$ in ${\mathrm{L}}({\mathcal{H}})$ such that

${\mathrm{wot}}\\!\\!\\!\lim_{n\to+\infty}U_{n}AU_{n}^{*}=X.$

Of course, we cannot replace the weak convergence by the strong convergence;
for instance if $A$ is invertible and $\|Xh\|<\|A^{-1}\|^{-1}$ for some unit
vector $h$, then $X$ cannot be a strong limit from the unitary orbit of $A$.
However, the next best thing does happen. Moreover, this is even true for the
$\ast$-strong operator topology.

######  Corollary 5.2.2.

Let $A,X\in{\mathrm{L}}({\mathcal{H}})$ with $W_{e}(A)\supset{\mathcal{D}}$
and $\|X\|\leq 1$. Then there exist two sequences of unitaries
$\\{U_{n}\\}_{n=1}^{\infty}$ and $\\{V_{n}\\}_{n=1}^{\infty}$ in
${\mathrm{L}}({\mathcal{H}})$ such that

$\ast\,{\mathrm{sot}}\\!\\!\\!\lim_{n\to+\infty}\frac{U_{n}AU_{n}^{*}+V_{n}AV_{n}^{*}}{2}=X.$

###### Proof.

From (5.1.1) we have

$A\simeq\begin{pmatrix}X&-R\\\ -S&T\end{pmatrix}.$

Hence there exist two unitaries
$U,V:{\mathcal{H}}\to{\mathcal{H}}\oplus{\mathcal{H}}$ such that

$\frac{UAU^{*}+VAV^{*}}{2}=\begin{pmatrix}X&0\\\ 0&T\end{pmatrix}.$ (5.2.1)

Now let $\\{e_{n}\\}_{n=1}^{\infty}$ be a basis of ${\mathcal{H}}$ and choose
any unitary $W_{n}:{\mathcal{H}}\oplus{\mathcal{H}}\to{\mathcal{H}}$ such that
$W_{n}(e_{j}\oplus 0)=e_{j}$ for all $j\leq n$. Then

$X_{n}:=W_{n}\begin{pmatrix}X&0\\\ 0&T\end{pmatrix}W_{n}^{*}$

strongly converges to $X$. Indeed, $\\{X_{n}\\}$ is bounded in norm and, for
all $j$, $X_{n}e_{j}\to Xe_{j}$. Taking adjoints,

$X_{n}^{*}=W_{n}\begin{pmatrix}X^{*}&0\\\ 0&T^{*}\end{pmatrix}W_{n}^{*},$

we also have $X_{n}^{*}\to X$ strongly. Setting $U_{n}=W_{n}U$ and
$V_{n}=W_{n}V$ and using (5.2.1) completes the proof. ∎

######  Remark 5.2.3.

Corollary 5.2.2 does not hold for the convergence in norm. We give an example.
Consider the permutation matrix

$T=\begin{pmatrix}0&0&1\\\ 1&0&0\\\ 0&1&0\end{pmatrix}$

and set $A=2(\oplus^{\infty}T)$ regarded as an operator in
${\mathrm{L}}({\mathcal{H}})$. Then $W_{e}(A)\supset{\mathcal{D}}$, however
$0$ cannot be a norm limit of means of two operators in the unitary orbit of
$A$. Indeed $0$ cannot be a norm limit of means of two operators in the
unitary orbit of $(A+A^{*})/2$ as $(A+A^{*})/2=2I-3P$ for some projection $P$.

######  Remark 5.2.4.

The converse of Corollary 5.2.2 holds: if $A\in{\mathrm{L}}({\mathcal{H}})$
has the property that any contraction is a strong limit of a mean of two
operators in its unitary orbit, then necessarily
$W_{e}(A)\supset{\mathcal{D}}$. This is checked by arguing as in the proof of
Corollary 5.4.1.

We reserve the word “projection” for self-adjoint idempotent. A strong limit
of uniformly bounded idempotent operators is still an idempotent; thus, the
next corollary is rather surprising.

######  Corollary 5.2.5.

Fix $\alpha>0$. There exists an idempotent $Q\in{\mathrm{L}}({\mathcal{H}})$
such that for every $X\in{\mathrm{L}}({\mathcal{H}})$ with $\|X\|\leq\alpha$
we have two sequences of unitaries $\\{U_{n}\\}_{n=1}^{\infty}$ and
$\\{V_{n}\\}_{n=1}^{\infty}$ in ${\mathrm{L}}({\mathcal{H}})$ for which

$\ast\,{\mathrm{sot}}\\!\\!\\!\lim_{n\to+\infty}U_{n}QU_{n}^{*}+V_{n}QV_{n}^{*}=X.$

###### Proof.

Let $a>0$, define a two-by-two idempotent matrix

$M_{a}=\begin{pmatrix}1&0\\\ a&0\end{pmatrix}$ (5.2.2)

and set $Q=\oplus^{\infty}M_{a}$ regarded as an operator in
${\mathrm{L}}({\mathcal{H}})$. Since the numerical range $W(\cdot)$ of

$\begin{pmatrix}0&0\\\ 2&0\end{pmatrix}$

is ${\mathcal{D}}$, we infer that
$W(2\alpha^{-1}M_{a})=W_{e}(2\alpha^{-1}Q)\supset{\mathcal{D}}$ for a large
enough $a$. The result then follows from Corollary 5.2.2 with
$A=2\alpha^{-1}Q$ and the contraction $\alpha^{-1}X$. ∎

Corollary 5.2.5 does not hold for the convergence in norm.

######  Proposition 5.2.6.

Let $X\in{\mathrm{L}}({\mathcal{H}})$ be of the form $\lambda I+K$ for a
compact operator $K$ and a scalar $\lambda\notin\\{0,1,2\\}$. Then $X$ is not
norm limit of $U_{n}QU_{n}^{*}+V_{n}QV_{n}^{*}$ for any sequences of unitaries
$\\{U_{n}\\}_{n=1}^{\infty}$ and $\\{V_{n}\\}_{n=1}^{\infty}$ and any
idempotent $Q$ in ${\mathrm{L}}({\mathcal{H}})$.

###### Proof.

First observe that if $\\{A_{n}\\}_{n=1}^{\infty}$ and
$\\{B_{n}\\}_{n=1}^{\infty}$ are two bounded sequences in
${\mathrm{L}}({\mathcal{H}})$ such that $A_{n}-B_{n}\to 0$ in norm, then we
also have $A_{n}^{2}-B_{n}^{2}\to 0$ in norm; indeed

$A_{n}^{2}-B_{n}^{2}=A_{n}(A_{n}-B_{n})+(A_{n}-B_{n})B_{n}.$

Now, suppose that $\lambda\neq 1$ and that we have the (norm) convergence,

$U_{n}QU_{n}^{*}+V_{n}QV_{n}^{*}\to\lambda I+K.$

Then we also have

$W_{n}QW_{n}^{*}-\left(-Q+\lambda I+U_{n}^{*}KU_{n}\right)\to 0$ (5.2.3)

where $W_{n}:=U_{n}^{*}V_{n}$. Hence, by the previous observation,

$(W_{n}QW_{n}^{*})^{2}-\left(-Q+\lambda I+U_{n}^{*}KU_{n}\right)^{2}\to 0,$

that is

$W_{n}QW_{n}^{*}-\left(-Q+\lambda I+U_{n}^{*}KU_{n}\right)^{2}\to 0.$ (5.2.4)

Combining (5.2.3) and (5.2.4) we get

$\left(-Q+\lambda I+U_{n}^{*}KU_{n}\right)-\left(-Q+\lambda
I+U_{n}^{*}KU_{n}\right)^{2}\to 0$

hence

$(-2+2\lambda)Q+(\lambda-\lambda^{2})I+K_{n}\to 0$

for some bounded sequence of compact operators $K_{n}$. Since $\lambda\neq 1$,
we have

$Q=\frac{\lambda}{2}I+L$

for some compact operator $L$. Since $Q$ is idempotent, either $\lambda=2$ or
$\lambda=0$. ∎

The operator $X$ in Proposition 5.2.6 has the special property that $W_{e}(X)$
is reduced to a single point. However Proposition 5.2.6 may also hold when
$W_{e}(X)$ has positive measure.

######  Corollary 5.2.7.

Let $Q$ be an idempotent in ${\mathrm{L}}({\mathcal{H}})$ and
$z\in\mathbb{C}\setminus\\{0,1,2\\}$. Then, there exists $\alpha>0$ such that
the following property holds:

  * If $X\in{\mathrm{L}}({\mathcal{H}})$ satisfies $\|X-zI\|\leq\alpha$, then $X$ is not norm limit of $U_{n}QU_{n}^{*}+V_{n}QV_{n}^{*}$ for any sequences of unitaries $\\{U_{n}\\}_{n=1}^{\infty}$ and $\\{V_{n}\\}_{n=1}^{\infty}$ in ${\mathrm{L}}({\mathcal{H}})$.

###### Proof.

By the contrary, $zI$ would be a norm limit of
$U_{n}QU_{n}^{*}+V_{n}QV_{n}^{*}$ for some unitaries $U_{n},V_{n}$,
contradicting Proposition 5.2.6. ∎

More operators with large numerical and essential numerical ranges are given
in the next proposition. An operator $X$ is stable when its real part
$(X+X^{*})/2$ is negative definite (invertible).

######  Proposition 5.2.8.

If $X\in{\mathrm{L}}({\mathcal{H}})$ is stable, then $X$ is not norm limit of
$U_{n}QU_{n}^{*}+V_{n}QV_{n}^{*}$ for any sequences of unitaries
$\\{U_{n}\\}_{n=1}^{\infty}$ and $\\{V_{n}\\}_{n=1}^{\infty}$ and any
idempotent $Q$ in ${\mathrm{L}}({\mathcal{H}})$.

###### Proof.

We have a decomposition
${\mathcal{H}}={\mathcal{H}}_{s}\oplus{\mathcal{H}}_{ns}$ in two invariant
subspaces of $Q$ such that $Q$ acts on ${\mathcal{H}}_{s}$ as a selfadjoint
projection $P$, and $Q$ acts on ${\mathcal{H}}_{ns}$ as a purely
nonselfadjoint idempotent, that is $A_{{\mathcal{H}}_{ns}}$ is unitarily
equivalent to an operator on ${\mathcal{F}}\oplus{\mathcal{F}}$ of the form

$Q_{{\mathcal{H}}_{ns}}\simeq\begin{pmatrix}I&0\\\ R&0\end{pmatrix}$ (5.2.5)

where $R$ is a nonsingular (i.e., a zero kernel) positive operator on a
Hilbert space ${\mathcal{F}}$, so

$Q\simeq P\oplus\begin{pmatrix}I&0\\\ R&0\end{pmatrix}.$ (5.2.6)

Let $Y$ be a norm limit of the sum of two sequences in the unitary orbit of
$Q$. If the purely non-selfadjoint part ${\mathcal{H}}_{ns}$ is vacuous, then
$Y$ is positive, hence $Y\neq X$. If ${\mathcal{H}}_{ns}$ is not vacuous,
(5.2.6) shows that

$\displaystyle Q+Q^{*}$ $\displaystyle\simeq 2P\oplus\begin{pmatrix}2I&R\\\
R&0\end{pmatrix}$ $\displaystyle\simeq 2P\oplus\left\\{\begin{pmatrix}I&I\\\
I&I\end{pmatrix}+\begin{pmatrix}R&0\\\ 0&-R\end{pmatrix}\right\\}.$

This implies that $\|(Q+Q^{*})_{+}\|\geq\|(Q+Q^{*})_{-}\|$, therefore
$Y+Y^{*}$ cannot be negative definite, hence $X\neq Y$. ∎

It is known [85] that any operator is the sum of five idempotents. We close
this section by asking whether Corollorary 5.2.5 admits a substitute for
Banach space operators.

######  Question 5.2.9.

Let ${\mathcal{X}}$ be a separable Banach space and
$T\in{\mathrm{L}}({\mathcal{X}})$, the linear operators on ${\mathcal{X}}$. Do
there exist two sequences $\\{P_{n}\\}_{n=1}^{\infty}$ and
$\\{Q_{n}\\}_{n=1}^{\infty}$ of idempotents in ${\mathrm{L}}({\mathcal{X}})$
such that $T={\mathrm{sot}}\lim_{n\to+\infty}(P_{n}+Q_{n})$ ?

####  5.3 Conditional expectation onto a masa

#####  5.3.1 Conditional expectation of general operators

Kennedy and Skoufranis have studied the following problem: Let
${\mathfrak{X}}$ be a maximal abelian $\ast$-subalgebra (masa) of a von
Neumann algebra ${\mathfrak{M}}$, with corresponding expectation
$\mathbb{E}_{\mathfrak{X}}:{\mathfrak{M}}\to{\mathfrak{X}}$ (i.e., a unital
positive linear map such that
$\mathbb{E}_{\mathfrak{X}}(XM)=X\mathbb{E}_{\mathfrak{X}}(M)$ for all
$X\in{\mathfrak{X}}$ and $M\in{\mathfrak{M}}$). Given a normal operator
$A\in{\mathfrak{M}}$, determine the image by $\mathbb{E}_{\mathfrak{X}}$ of
the unitary orbit of $A$,

$\Delta_{\mathfrak{X}}(A)=\\{\,\mathbb{E}_{\mathfrak{X}}(UAU^{*})\ :\
U\,{\mathrm{a\ unitary\ in}}\ {\mathfrak{M}}\,\\}.$

In several cases, they determined the norm closure of
$\Delta_{\mathfrak{X}}(A)$. In particular, [73, Theorem 1.2] can be stated in
the following two propositions.

######  Proposition 5.3.1.

Let ${\mathfrak{X}}$ be a masa in ${\mathrm{L}}({\mathcal{H}})$,
$X\in{\mathfrak{X}}$, and $A$ a normal operator in
${\mathrm{L}}({\mathcal{H}})$. If
$\sigma(X)\subset{\mathrm{conv}}\sigma_{e}(A)$, then $X$ lies in the norm
closure of $\Delta_{\mathfrak{X}}(A)$.

######  Proposition 5.3.2.

Let ${\mathfrak{X}}$ be a continuous masa in ${\mathrm{L}}({\mathcal{H}})$,
$X\in{\mathfrak{X}}$, and $A$ a normal operator in
${\mathrm{L}}({\mathcal{H}})$. If $X$ lies in the norm closure of
$\Delta_{\mathfrak{X}}(A)$, then
$\sigma(X)\subset{\mathrm{conv}}\sigma_{e}(A)$.

Since we deal with normal operators,
$\sigma(X)\subset{\mathrm{conv}}\sigma_{e}(A)$ means $W(X)\subset W_{e}(A)$.
Proposition 5.3.2 needs the continuous assumption. It is a rather simple fact;
we generalize it in Lemma 5.3.5: Conditional expectations reduce essential
numerical ranges, $W_{e}(\mathbb{E}_{\mathfrak{X}}(T))\subset W_{e}(T)$ for
all $T\in{\mathrm{L}}({\mathcal{H}})$. Thus, the main point of [73, Theorem
1.2] is Proposition 5.3.1 which says that if $W(X)\subset W_{e}(A)$ then $X$
can be approximated by operators of the form
$\mathbb{E}_{\mathfrak{X}}(UAU^{*})$ with unitaries $U$. With the slightly
stronger assumption $W(X)\subset_{st}W_{e}(A)$, Theorem 5.1.2 guarantees, via
the following corollary, that $X$ is exactly of this form. Furthermore the
normality assumption on $A$ is not necessary.

######  Corollary 5.3.3.

Let ${\mathfrak{X}}$ be a masa in ${\mathrm{L}}({\mathcal{H}})$,
$X\in{\mathfrak{X}}$ and $A\in{\mathrm{L}}({\mathcal{H}})$. If
$W(X)\subset_{st}W_{e}(A)$, then $X=\mathbb{E}_{\mathfrak{X}}(UAU^{*})$ for
some unitary operator $U\in{\mathrm{L}}({\mathcal{H}})$.

###### Proof.

First, we note a simple fact: Let $\\{P_{i}\\}_{i=1}^{\infty}$ be a sequence
of orthogonal projections in ${\mathfrak{X}}$ such that
$\sum_{i=1}^{\infty}P_{i}=I$, and let $Z\in{\mathrm{L}}({\mathcal{H}})$ such
that $P_{i}ZP_{i}\in{\mathfrak{X}}$ for all $i$. Then, we have a strong sum

$\mathbb{E}_{\mathfrak{X}}(Z)=\sum_{i=1}^{\infty}P_{i}ZP_{i}.$

Now, denote by ${\mathcal{H}}_{i}$ the range of $P_{i}$ and assume
$\dim{\mathcal{H}}_{i}=\infty$ for all $i$. We have
$W(X_{{\mathcal{H}}_{i}})\subset W(X)$, hence

$\cup_{i=1}^{\infty}W(X_{{\mathcal{H}}_{i}})\subset_{st}W_{e}(A).$

We may then apply Theorem 5.1.2 and get a unitary $U$ on
${\mathcal{H}}=\bigoplus_{i=1}^{\infty}{\mathcal{H}}_{i}$ such that

$A\simeq UAU^{*}=\begin{pmatrix}X_{{\mathcal{H}}_{1}}&\ast&\cdots&\cdots\\\
\ast&X_{{\mathcal{H}}_{2}}&\ast&\cdots\\\ \vdots&\ast&\ddots&\ddots\\\
\vdots&\vdots&\ddots&\ddots\\\ \end{pmatrix}.$

Since $0\oplus\cdots\oplus X_{{\mathcal{H}}_{i}}\oplus
0\cdots\in{\mathfrak{X}}$ for all $i$, the previous simple fact shows that

$\mathbb{E}_{\mathfrak{X}}(UAU^{*})=\bigoplus_{i=1}^{\infty}X_{{\mathcal{H}}_{i}}=X.$

∎

We remark that Corollary 5.3.3 also covers the assumption $W(X)\subset
W_{e}(A)$ of Proposition 5.3.1. Indeed, $W_{e}(A)\subset_{st}W_{e}(A+D)$ for
some normal operator $D$ with arbitrarily small norm, and we may apply
Corollary 5.3.3 to $X$ and $A+D$.

Kadison’s article [71] completely describes the diagonals of a projection,
thus, providing in his terminology a carpenter theorem for discrete masas
${\mathrm{L}}({\mathcal{H}})$. Our method can be used to obtain a similar
statement for continuous masas in ${\mathrm{L}}({\mathcal{H}})$ given in the
next corollary. This result is due to Akemann and Anderson, see [1, Corollary
6.19]. The case of a masa in a type-${\mathrm{II}}_{1}$ factor is solved in
Ravichandran’s paper [87].

######  Corollary 5.3.4.

Let ${\mathfrak{X}}$ be a continuous masa in ${\mathrm{L}}({\mathcal{H}})$ and
let $X\in{\mathfrak{X}}$ be a positive contraction. Then there exists a
projection $P$ in ${\mathrm{L}}({\mathcal{H}})$ such that
$X=\mathbb{E}_{\mathfrak{X}}(P)$.

###### Proof.

We may suppose that $0<\|Xh\|<1$ for all unit vectors $h\in{\mathcal{H}}$;
otherwise decompose $X$ as $Q+X_{0}$ for some projection $Q\in{\mathfrak{X}}$
and consider $X_{0}$ in place of $X$. Then we have a decomposition
${\mathcal{H}}=\bigoplus_{i=1}^{\infty}{\mathcal{H}}_{i}$, where each summand
${\mathcal{H}}_{i}$ is the range of some projection in ${\mathfrak{X}}$, such
that the decomposition $X=\bigoplus_{i=1}^{\infty}X_{{\mathcal{H}}_{i}}$
satisfies $W(X_{{\mathcal{H}}_{i}})\subset_{st}[0,1]$ for all $i$. We may
further decompose each Hilbert space ${\mathcal{H}}_{i}$ as
${\mathcal{H}}_{i}=\bigoplus_{j=1}^{\infty}{\mathcal{H}}_{i,j}$ where
${\mathcal{H}}_{i,j}$ is the range of some nonzero projection in
${\mathfrak{X}}$. We have a corresponding decomposition
$X=\bigoplus_{i=1}^{\infty}\left(\bigoplus_{j=1}^{\infty}X_{{\mathcal{H}}_{i,j}}\right)$
Since $\cup_{j=1}^{\infty}W(X_{{\mathcal{H}}_{i,j}})\subset
W(X_{{\mathcal{H}}_{i}})\subset_{st}[0,1]$, Corollary 5.1.3 yields some
projection $P_{i}\in{\mathrm{L}}({\mathcal{H}}_{i})$, $W_{e}(P)=[0,1]$, such
that

$P_{i}=\begin{pmatrix}X_{{\mathcal{H}}_{i,1}}&\ast&\cdots&\cdots\\\
\ast&X_{{\mathcal{H}}_{i,2}}&\ast&\cdots\\\ \vdots&\ast&\ddots&\ddots\\\
\vdots&\vdots&\ddots&\ddots\\\ \end{pmatrix}.$

Letting $P=\bigoplus_{i=1}^{\infty}P_{i}$, we have
$X=\mathbb{E}_{\mathfrak{X}}(P)$. ∎

#####  5.3.2 A reduction lemma

The following result extends Proposition 5.3.2, the “easy” part of Kennedy-
Skoufranis’ theorem [73, Theorem 1.2].

######  Lemma 5.3.5.

If ${\mathfrak{X}}$ is a masa in ${\mathrm{L}}({\mathcal{H}})$ and
$Z\in{\mathrm{L}}({\mathcal{H}})$, then
$W(\mathbb{E}_{\mathfrak{X}}(Z))\subset\overline{W}(Z)$ and
$W_{e}(\mathbb{E}_{\mathfrak{X}}(Z))\subset W_{e}(Z)$.

###### Proof.

(1) Assume $Z$ is normal. We may identify the unital $C^{*}$-algebra
${\mathfrak{A}}$ spanned by $Z$ with $C^{0}(\sigma(Z))$ via a
$\ast$-isomorphism $\varphi:C^{0}(\sigma(Z))\to{\mathfrak{A}}$ with
$\varphi(z\mapsto z)=Z$. Let $h\in{\mathcal{H}}$ be a unit vector. For $f\in
C^{0}(\sigma(Z))$, set

$\psi(f)=\langle h,\mathbb{E}_{\mathfrak{X}}(\varphi(f))h\rangle.$

Then $\psi$ is a positive linear functional on $C^{0}(\sigma(Z))$ and
$\psi(1)=1$. Thus $\psi$ is a Radon measure induced by a probabilty measure
$\mu$,

$\psi(f)=\int_{\sigma(Z)}f(z)\,{\mathrm{d}}\mu(z).$

We then have $\langle
h,\mathbb{E}_{\mathfrak{X}}(Z)h\rangle=\psi(z)\in{\mathrm{conv}}(\sigma(Z)).$
Since ${\mathrm{conv}}(\sigma(Z))=\overline{W}(Z)$, we obtain
$W(\mathbb{E}_{\mathfrak{X}}(Z))\subset\overline{W}(Z)$.

(2) Let $Z$ be a general operator in ${\mathrm{L}}({\mathcal{H}})$ and define
a conditional expectation

$\mathbb{E}_{2}:{\mathrm{L}}({\mathcal{H}}\oplus{\mathcal{H}})\to{\mathfrak{X}}\oplus{\mathfrak{X}}$

by

$\mathbb{E}_{2}\left(\begin{pmatrix}A&C\\\
D&B\end{pmatrix}\right)=\begin{pmatrix}\mathbb{E}_{\mathfrak{X}}(A)&0\\\
0&\mathbb{E}_{\mathfrak{X}}(B)\end{pmatrix}.$

From the first part of the proof, we infer

$W(\mathbb{E}_{\mathfrak{X}}(Z))\subset
W\left(\begin{pmatrix}\mathbb{E}_{\mathfrak{X}}(Z)&0\\\
0&\mathbb{E}_{\mathfrak{X}}(B)\end{pmatrix}\right)\subset\overline{W}\left(\begin{pmatrix}Z&C\\\
D&B\end{pmatrix}\right)$

whenever $\begin{pmatrix}Z&C\\\ D&B\end{pmatrix}$ is normal. Since we have, by
a simple classical fact [60],

$\overline{W}(Z)=\bigcap\overline{W}\left(\begin{pmatrix}Z&C\\\
D&B\end{pmatrix}\right)$

where the intersection runs over all $B,C,D$ such that $\begin{pmatrix}Z&C\\\
D&B\end{pmatrix}$ is normal, we obtain
$W(\mathbb{E}_{\mathfrak{X}}(Z))\subset\overline{W}(Z)$.

(3) We deal with the essential numerical range inclusion. We can split
${\mathfrak{X}}$ into its discrete part ${\mathfrak{D}}$ and continuous part
${\mathfrak{C}}$ with the corresponding decomposition of the Hilbert space,

${\mathfrak{X}}={\mathfrak{D}}\oplus{\mathfrak{C}},\qquad{\mathcal{H}}={\mathcal{H}}_{d}\oplus{\mathcal{H}}_{c}.$

We then have

$W_{e}(\mathbb{E}_{\mathfrak{X}}(Z))={\mathrm{conv}}\left\\{W_{e}(\mathbb{E}_{\mathfrak{D}}(Z_{{\mathcal{H}}_{d}}));W_{e}(\mathbb{E}_{\mathfrak{C}}(Z_{{\mathcal{H}}_{c}}))\right\\}.$
(5.3.1)

We have an obvious inclusion

$W_{e}(\mathbb{E}_{\mathfrak{D}}(Z_{{\mathcal{H}}_{d}}))\subset
W_{e}(Z_{{\mathcal{H}}_{d}}).$ (5.3.2)

On the other hand, for all compact operators
$K\in{\mathrm{L}}({\mathcal{H}})$,

$W_{e}(\mathbb{E}_{\mathfrak{C}}(Z_{{\mathcal{H}}_{c}}))=W_{e}(\mathbb{E}_{\mathfrak{C}}(Z_{{\mathcal{H}}_{c}})+K_{{\mathcal{H}}_{c}})=W_{e}(\mathbb{E}_{\mathfrak{C}}(Z_{{\mathcal{H}}_{c}}+K_{{\mathcal{H}}_{c}}))\subset\overline{W}(Z_{{\mathcal{H}}_{c}}+K_{{\mathcal{H}}_{c}})$

by the simple folklore fact that a conditional expectation onto a continous
masa vanishes on compact operators and part (2) of the proof. Thus, when $K$
runs over all compact operators, we obtain

$W_{e}(\mathbb{E}_{\mathfrak{D}}(Z_{{\mathcal{H}}_{c}}))\subset
W_{e}(Z_{{\mathcal{H}}_{c}}).$ (5.3.3)

Combining (5.3.1), (5.3.2) and (5.3.3) completes the proof. ∎

#####  5.3.3 Conditional expectation of idempotent operators

For discrete masas, unlike continuous masas [72], there is a unique
conditional expectation, which merely consists in extracting the diagonal with
respect to an orthonormal basis. In a recent article, Loreaux and Weiss give a
detailed study of diagonals of idempotents in ${\mathrm{L}}({\mathcal{H}})$.
They established that a nonzero idempotent $Q$ has a zero diagonal with
respect to some orthonormal basis if and only if $Q$ is not a Hilbert-Schmidt
perturbation of a projection (i.e., a self-adjoint idempotent). They also
showed that any sequence $\\{a_{n}\\}\in l^{\infty}$ such that
$|a_{n}|\leq\alpha$ for all $n$ and, for some $a_{n_{0}}$, $a_{k}=a_{n_{0}}$
for infinitely many $k$, one has an idempotent $Q$ such that $\|Q\|\leq
18\alpha+4$ and $Q$ admits $\\{a_{n}\\}$ as a diagonal with respect to some
orthonormal basis [78, Proposition 3.4]. Using this, they proved that any
sequence in $l^{\infty}$ is the diagonal of some idempotent operator [78,
Theorem 3.6], answering a question of Jasper. This statement is in the range
of Theorem 5.1.1. Further, it is not necessary to confine to diagonals, i.e.,
discrete masas, and the constant $18\alpha+4$ can be improved; in the next
corollary we have $3$ as the best constant when $\alpha=1$.

######  Corollary 5.3.6.

Let ${\mathfrak{X}}$ be a masa in ${\mathrm{L}}({\mathcal{H}})$ and
$\alpha>0$. There exists an idempotent $Q\in{\mathrm{L}}({\mathcal{H}})$, such
that for all $X\in{\mathfrak{X}}$ with $\|X\|<\alpha$, we have
$X=\mathbb{E}_{\mathfrak{X}}(UQU^{*})$ for some unitary operator
$U\in{\mathrm{L}}({\mathcal{H}})$. If $\alpha=1$, $\|Q\|=3$ is the smallest
possible norm.

###### Proof.

As in the proof of Corollary 5.2.5 we have an idempotent $Q$ such that
$W_{e}(Q)\supset\alpha{\mathcal{D}}$, hence the first and main part of
Corollary 5.3.6 follows from Corollary 5.3.3. The remaining parts require a
few computations.

To obtain the bound $3$ when $\alpha=1$ we get a closer look at
$\oplus^{\infty}M_{a}$ with $M_{a}$ given by (5.2.2) where $a$ is a positive
scalar. We have

$\displaystyle W(M_{a})$ $\displaystyle=\left\\{\langle h,M_{a}h\rangle\ :\
h\in\mathbb{C}^{2},\|h\|=1\right\\}$
$\displaystyle=\left\\{|h_{1}|^{2}+a\overline{h_{2}}h_{1}\
:|h_{1}|^{2}+|h_{2}|^{2}=1\right\\},$

hence, with $h_{1}=re^{i\theta}$, $h_{2}=\sqrt{1-r^{2}}e^{i\alpha}$,

$W(M_{a})=\bigcup_{0\leq r\leq
1}\left\\{r^{2}+ar\sqrt{1-r^{2}}e^{i(\theta-\alpha)}:\
\theta,\alpha\in[0,2\pi]\right\\}.$

Therefore $W(M_{a})$ is a union of circles $\Gamma_{r}$ with centers $r^{2}$
and radii $ar\sqrt{1-r^{2}}$. To have ${\mathcal{D}}\subset W(M_{a})$ it is
necessary and sufficient that $-1\in\Gamma_{r}$ for some $r\in[0,1]$, hence

$a=\frac{1+r^{2}}{r\sqrt{1-r^{2}}}.$ (5.3.4)

Now we minimize $a=a(r)$ given by (5.3.4) when $r\in(0,1)$ and thus obtain the
matrix $M_{a_{\ast}}$ with smallest norm such that
$W(M_{a_{\ast}})\supset{\mathcal{D}}$. Observe that $a(r)\to+\infty$ as $r\to
0$ and as $r\to 1$, and

$r^{2}(1-r^{2})^{3/2}a^{\prime}(r)=3r^{2}-1.$

Thus $a(r)$ takes its minimal value $a_{\ast}$ when $r=1/\sqrt{3}$. We have
$a_{\ast}=2\sqrt{2}$, hence

$\|M_{a_{\ast}}\|=3.$

Now, letting $Q=\oplus^{\infty}M_{a_{\ast}}$, we have
$W_{e}(Q)=W(M_{a_{\ast}})$, so that $Q$ is an idempotent in
${\mathrm{L}}({\mathcal{H}})$ such that $W_{e}(Q)\supset{\mathcal{D}}$, and
thus by Corollary 5.3.3 any operator $X$ such that $\|X\|<1$ satifies
$\mathbb{E}_{\mathfrak{X}}(UQU^{*})=X$ for some unitary $U$.

It remains to check that if $Q$ is an idempotent such that Corollary 5.3.6
holds for any operator $X$ such that $\|X\|<1$, then $\|Q\|\geq 3$. To this
end, we consider the purely nonselfadjoint part $Q_{{\mathcal{H}}_{ns}}$ of
$Q$ in (5.2.5),

$Q_{{\mathcal{H}}_{ns}}\simeq\begin{pmatrix}I&0\\\ R&0\end{pmatrix}.$

We have $W_{e}(Q)\supset{\mathcal{D}}$ if and only if
$W_{e}(Q_{{\mathcal{H}}_{ns}})\supset{\mathcal{D}}$. By Lemma 5.3.5 this is
necessary. We may approximate $W_{e}(Q_{{\mathcal{H}}_{ns}})$ with sligthly
larger essential numerical ranges, by using a positive diagonalizable operator
$R_{\varepsilon}$ such that $R_{\varepsilon}\geq R\geq
R_{\varepsilon}-\varepsilon I$, for which

$W_{e}\left(\begin{pmatrix}I&0\\\
R_{\varepsilon}&0\end{pmatrix}\right)=W_{e}\left(\bigoplus_{n=1}^{\infty}\begin{pmatrix}1&0\\\
a_{n}&0\end{pmatrix}\right)$

where $\\{a_{n}\\}_{n=1}^{\infty}$ is a sequence of positive scalars, the
eigenvalues of $R_{\varepsilon}$. By the previous step of the proof, this
essential numerical range contains $\mathcal{D}$ if and only if
$\overline{\lim}\,a_{n}\geq a_{\ast}$. If this holds for all $\varepsilon>0$,
then $\|Q\|\geq 3$. ∎

####  5.4 Unital, trace preserving positive linear maps

Unital positive linear maps $\Phi:\mathbb{M}_{n}\to\mathbb{M}_{n}$, the matrix
algebra, which preserve the trace play an important role in matrix analysis
and its applications. These maps are sometimes called doubly stochastic [5].

We say that
$\Phi:{\mathrm{L}}({\mathcal{H}})\mapsto{\mathrm{L}}({\mathcal{H}})$ is trace
preserving if it preserves the trace ideal ${\mathcal{T}}$ and
${\mathrm{Tr}}\,\Phi(Z)={\mathrm{Tr}}\,Z$ for all $Z\in{\mathcal{T}}$.

######  Corollary 5.4.1.

Let $A\in{\mathrm{L}}({\mathcal{H}})$. The following two conditions are
equivalent:

  * (i)

$W_{e}(A)\supset{\mathcal{D}}$.

  * (ii)

For all $X\in{\mathrm{L}}({\mathcal{H}})$ with $\|X\|<1$, there exists a
unital, trace preserving, positive linear map
$\Phi:{\mathrm{L}}({\mathcal{H}})\to{\mathrm{L}}({\mathcal{H}})$ such that
$\Phi(A)=X$.

We may further require in (ii) that $\Phi$ is completely positive and sot\-
and wot-sequentially continuous.

###### Proof.

Assume (i). By Theorem 5.1.1 we have a unitary
$U:{\mathcal{H}}\to\oplus^{\infty}{\mathcal{H}}$ such that

$A\simeq UAU^{*}=\begin{pmatrix}X&\ast&\cdots&\cdots\\\ \ast&X&\ast&\cdots\\\
\vdots&\ast&X&\ddots\\\ \vdots&\vdots&\ddots&\ddots\\\ \end{pmatrix}.$

Now consider the map
$\Psi:{\mathrm{L}}(\oplus^{\infty}{\mathcal{H}})\to{\mathrm{L}}({\mathcal{H}})$,

$\begin{pmatrix}Z_{1,1}&Z_{1,2}&\cdots\\\ Z_{2,1}&Z_{2,2}&\cdots\\\
\vdots&\vdots&\ddots\\\ \end{pmatrix}\mapsto\sum_{i=1}^{\infty}2^{-i}Z_{i,i}$

and define $\Phi:{\mathrm{L}}({\mathcal{H}})\to{\mathrm{L}}({\mathcal{H}})$ as
$\Phi(T)=\Psi(UTU^{*})$. Since both $\Psi$ and the unitary congruence with $U$
are sot- and wot-sequentially continuous, and trace preseverving, completely
positive and unital, so is $\Phi$. Further $\Phi(A)=X$.

Assume (ii) and suppose that $z\notin W_{e}(A)$ and $|z|<1$ in order to reach
a contradiction. If $z=|z|e^{i\theta}$, replacing $A$ by $e^{-i\theta}A$, we
may assume $1>z\geq 0$. Hence,

$W_{e}((A+A^{*})/2)\subset(-\infty,z]$

and there exists a selfadjoint compact operator $L$ such that

$\frac{A+A^{*}}{2}\leq zI+L.$

This implies that $X:=\frac{1+z}{2}I$ cannot be in the range of $\Phi$ for any
unital, trace preserving positive linear map. Indeed, we would have

$\frac{1+z}{2}I=\frac{X+X^{*}}{2}=\Phi\left(\frac{A+A^{*}}{2}\right)\leq
zI+\Phi(L)$

which is not possible as $\Phi(L)$ is compact. ∎

In the finite dimensional setting, two Hermitian matrices $A$ and $X$ satisfy
the relation $X=\Phi(A)$ for some positive, unital, trace preserving linear
map if and only if $X$ is in the convex hull of the unitary orbit of $A$. In
the infinite dimensional setting, if two Hermitian
$A,X\in{\mathrm{L}}({\mathcal{H}})$ satisfy $W_{e}(A)\supset[-1,1]$ and
$\|X\|\leq 1$, then $X$ is in the norm closure of the unitary orbit of $A$.
This is easily checked by approximating the operators with diagonal operators.
Such an equivalence might not be brought out to the setting of Corollary
5.4.1.

######  Question 5.4.2.

Do there exist $A,X\in{\mathrm{L}}({\mathcal{H}})$ such that
$W_{e}(A)\supset{\mathcal{D}}$, $\|X\|<1$, and $X$ does not belong to the norm
closure of the convex hull of the unitary orbit of $A$ ?

Here we mention a result of Wu [93, Theorem 6.11]: If
$A\in{\mathrm{L}}({\mathcal{H}})$ is not of the form scalar plus compact, then
every $X\in{\mathrm{L}}({\mathcal{H}})$ is a linear combination of operators
in the unitary orbit of $A$.

If one deletes the positivity assumption, the most regular class of linear
maps on ${\mathrm{L}}({\mathcal{H}})$ might be given in the following
definition.

######  Definition 5.4.3.

A linear map $\Psi:{\mathrm{L}}({\mathcal{H}})\to{\mathrm{L}}({\mathcal{H}})$
is said ultra-regular if it fulfills two conditions:

  * (u1)

$\Psi(I)=I$ and $\Psi$ is trace preserving.

  * (u2)

Whenever a sequence $A_{n}\to A$ for either the norm-, strong-, or weak-
topology, then we also have $\Psi(A_{n})\to\Psi(A)$ for the same type of
convergence.

Any ultra-regular linear map preserves the set of essentially scalar operators
(of the form $\lambda I+K$ with $\lambda\in\mathbb{C}$ and a compact operator
$K$). For its complement, we state our last corollary.

######  Corollary 5.4.4.

Let $A\in{\mathrm{L}}({\mathcal{H}})$ be essentially nonscalar. Then, for all
$X\in{\mathrm{L}}({\mathcal{H}})$ there exists a ultra-regular linear map
$\Psi:{\mathrm{L}}({\mathcal{H}})\to{\mathrm{L}}({\mathcal{H}})$ such that
$\Psi(A)=X$.

###### Proof.

An operator is essentially nonscalar precisely when its essential numerical
range is not reduced to a single point. So, let $a,b\in W_{e}(A)$, $a\neq b$.
By a lemma of Anderson and Stampfli [3], $A$ is unitarily equivalent to an
operator on ${\mathcal{H}}\oplus{\mathcal{H}}$ of the form

$B=\begin{pmatrix}D&\ast\\\ \ast&\ast\end{pmatrix}$

where $D=\oplus_{n=1}^{\infty}D_{n}$, with two by two matrices $D_{n}$,

$D_{n}=\begin{pmatrix}a_{n}&0\\\ 0&b_{n}\end{pmatrix}$

such that $a_{n}\to a$ and $b_{n}\to b$ as $n\to\infty$. We may assume that,
for some $\alpha,\beta>0$, we have $\alpha>|a_{n}|+|b_{n}|$ and
$|a_{n}-b_{n}|>\beta$. Hence there exist $\gamma>0$ and two by two intertible
matrices $T_{n}$ such that, for all $n$,
$W(T_{n}D_{n}T_{n}^{-1})\supset{\mathcal{D}}$ and
$\|T_{n}\|+\|T_{n}^{-1}\|\leq\gamma$. So, letting
$T=\left(\oplus_{n=1}^{\infty}T_{n}\right)\oplus I$, we obtain an invertible
operator $T$ on ${\mathcal{H}}\oplus{\mathcal{H}}$ such that
$W_{e}(TBT^{-1})\supset{\mathcal{D}}$.

Hence we have an invertible operator $S$ on ${\mathcal{H}}$ such that
$W_{e}(SAS^{-1})\supset{\mathcal{D}}$. Therefore we may apply Corollary 5.4.1
and obtain a wot- and sot-sequentially continuous, unital, trace preserving
map $\Phi$ such that $\Phi(SAS^{-1})=X$. Letting $\Psi(\cdot)=\Phi(S\cdot
S^{-1})$ completes the proof. ∎

We cannot find an alternative proof, not based on the pinching theorem, for
Corollaries 5.4.1 and 5.4.4.

If we trust in Zorn, there exists a linear map
$\Psi:{\mathrm{L}}({\mathcal{H}})\to{\mathrm{L}}({\mathcal{H}})$ which
satifies the condition (u1) but not the condition (u2). Indeed, let
$\\{a_{p}\\}_{p\in\Omega}$ be a basis in the Calkin algebra
${\mathfrak{C}}={\mathrm{L}}({\mathcal{H}})/{\mathrm{K}}({\mathcal{H}})$,
indexed on an ordered set $\Omega$, whose first element $a_{p_{0}}$ is the
image of $I$ by the canonical projection
$\pi:{\mathrm{L}}({\mathcal{H}})\to{\mathfrak{C}}$. Thus, for each operator
$X$, we have a unique decomposition
$\pi(X)=\sum_{p\in\Omega}(\pi(X))_{p}a_{p}$ with only finitely many nonzero
terms. Further $(\pi(X))_{p_{0}}=0$ if $X$ is compact, and
$(\pi(I))_{p_{0}}=1$. We then define a map
$\psi:{\mathrm{L}}({\mathcal{H}})\to{\mathrm{L}}({\mathcal{H}}\oplus{\mathcal{H}})$
by

$\psi(X)=\begin{pmatrix}X&0\\\ 0&(\pi(X))_{p_{0}}I\end{pmatrix}.$

Letting $\Psi(X)=V\psi(X)V^{*}$ where
$V:{\mathcal{H}}\oplus{\mathcal{H}}\to{\mathcal{H}}$ is unitary, we obtain a
linear map $\Psi:{\mathrm{L}}({\mathcal{H}})\to{\mathrm{L}}({\mathcal{H}})$
which satifies (u1) but not (u2): it is not norm continuous.

Let $\omega$ be a Banach limit on $l^{\infty}$ and define a map
$\phi:l^{\infty}\to l^{\infty}$, $\\{a_{n}\\}\mapsto\\{b_{n}\\}$, where
$b_{1}=\omega(\\{a_{n}\\})$ and $b_{n}=a_{n-1}$, $n\geq 2$. Letting
$\Psi(X)=\phi(diag(X))$, where $diag(X)$ is the diagonal of
$X\in{\mathcal{H}}$ in an orthonormal basis, we obtain a linear map $\Psi$
which is norm continuous, satisfies (u1) but not (u2): it is not strongly
sequentially continuous.

However, it seems not possible to define explicitly a linear map
$\Psi:{\mathrm{L}}({\mathcal{H}})\to{\mathrm{L}}({\mathcal{H}})$ satisfying
(u1) but not (u2).

####  5.5 Pinchings in factors ?

We discuss possible extensions to our results to a von Neumann algebra
${\mathfrak{R}}$ acting on a separable Hilbert space ${\mathcal{H}}$. First,
we need to define an essential numerical range $W_{e}^{\mathfrak{R}}$ for
${\mathfrak{R}}$. Let $A\in{\mathfrak{R}}$. If ${\mathfrak{R}}$ is
type-${\mathrm{III}}$, then $W_{e}^{\mathfrak{R}}(A):=W_{e}(A)$. If
${\mathfrak{R}}$ is type-${\mathrm{II}}_{\infty}$, then

$W_{e}^{\mathfrak{R}}(A):=\bigcap_{K\in{\mathcal{T}}}\overline{W}(A+K)$

where ${\mathcal{T}}$ is the trace ideal in ${\mathfrak{R}}$ (we may also use
its norm closure ${\mathcal{K}}$, the “compact” operators in ${\mathfrak{R}}$,
or any dense sequence in ${\mathcal{K}}$)

######  Question 5.5.1.

In Corollaries 5.2.1, 5.2.2 and 5.2.5, can we replace
${\mathrm{L}}({\mathcal{H}})$ by a type-${\mathrm{II}}_{\infty}$ or
-${\mathrm{III}}$ factor ${\mathfrak{R}}$ with $W_{e}^{\mathfrak{R}}$ ?

######  Question 5.5.2.

In Corollaries 5.3.3 and 5.3.6, can we replace ${\mathrm{L}}({\mathcal{H}})$
by a type-${\mathrm{II}}_{\infty}$ or -${\mathrm{III}}$ factor
${\mathfrak{R}}$ with $W_{e}^{\mathfrak{R}}$ ?

######  Question 5.5.3.

In Corollaries 5.4.1 and 5.4.4, can we replace ${\mathrm{L}}({\mathcal{H}})$
by a type-${\mathrm{II}}_{\infty}$ factor ${\mathfrak{R}}$ with
$W_{e}^{\mathfrak{R}}$ ?

Recently, Dragan and Kaftal [58] obtained some decompositions for positive
operators in von Neumann factors, which, in the case of
${\mathrm{L}}({\mathcal{H}})$ were first investigated in [33]-[34] by using
Theorem 5.1.1. This suggests that our questions dealing with a possible
extension to type-${\mathrm{II}}_{\infty}$ and -${\mathrm{III}}$ factors also
have an affirmative answer. In fact, it seems pausible that Theorem 5.1.1 and
Theorem 5.1.2 admit a version for such factors and this would affirmatively
answer these questions.

Let ${\mathfrak{R}}$ be a type-${\mathrm{II}}_{\infty}$ or -${\mathrm{III}}$
factor.

######  Definition 5.5.4.

A sequence $\\{V_{i}\\}_{i=1}^{\infty}$ of isometries in ${\mathfrak{R}}$ such
that $\sum_{i=1}^{\infty}V_{i}V_{i}^{*}=I$ is called an isometric
decomposition of ${\mathfrak{R}}$.

######  Conjecture 5.5.5.

Let $A\in{\mathfrak{R}}$ with $W_{e}^{\mathfrak{R}}(A)\supset{\mathcal{D}}$
and $\\{X_{i}\\}_{i=1}^{\infty}$ a sequence in ${\mathfrak{R}}$ such that
$\sup_{i}\|X_{i}\|<1$. Then, there exists an isometric decomposition
$\\{V_{i}\\}_{i=1}^{\infty}$ of ${\mathfrak{R}}$ such that
$V_{i}^{*}AV_{i}=X_{i}$ for all $i$.

####  5.6 Around this article

#####  5.6.1 Essential numerical range

Let us give three equivalent definitions of the essential numerical range
$W_{e}(A)$ of an operator $A$ acting on the Hilbert space ${\mathcal{H}}$.

  * (1)

$W_{e}(A)=\cap\overline{W}(A+K)$, the intersection running over the compact
operators $K$

  * (2)

Let $\\{E_{n}\\}$ be any sequence of finite rank projections converging
strongly to the identity and denote by $B_{n}$ the compression of $A$ to the
subspace $E_{n}^{\perp}$. Then $W_{e}(A)=\cap_{n\geq 1}\overline{W}(B_{n})$

  * (3)

$W_{e}(A)=\\{\lambda\ |\ {\rm there\ is\ an\ orthonormal\ system\
\\{e_{n}\\}_{n=1}^{\infty}\ with\ \lim\langle
e_{n},Ae_{n}\rangle=\lambda}\\}.$

It follows that $W_{e}(A)$ is a compact convex set containing the essential
spectrum of $A$, $Sp_{e}(A)$. The equivalence between these definitions has
been known since the early seventies if not soone. The very first definition
of $W_{e}(A)=$ is (1); however (3) is also a natural notion and easily entails
convexity and compactness of the essential numerical range.

#####  5.6.2 Proof of the pinching theorem

Recall that an operator mean an element of the algebra ${\rm L}({\cal H})$ of
all bounded linear operators acting on the usual (i.e. complex, separable,
infinite dimensional) Hilbert space ${\cal H}$. We will denote by the same
letter a projection and the corresponding subspace. Thus, if $F$ is a
projection and $A$ is an operator, we denote by $A_{F}$ the compression of $A$
by $F$, that is the restriction of $FAF$ to the subspace $F$. Given a total
sequence of nonzero mutually orthogonal projections $\\{E_{n}\\}$, we consider
the pinching

${\cal
P}(A)=\sum_{n=1}^{\infty}E_{n}AE_{n}=\bigoplus_{n=1}^{\infty}A_{E_{n}}.$

If $\\{A_{n}\\}$ is a sequence of operators acting on separable Hilbert spaces
with $A_{n}$ unitarily equivalent to $A_{E_{n}}$ for all $n$, we also
naturally write ${\cal P}(A)\simeq\bigoplus_{n=1}^{\infty}A_{n}$. Our main
result [21] for operator diagonals can then be stated as:

######  Theorem 5.6.1.

Let $A$ be an operator with $W_{e}(A)\supset{\cal D}$ and let
$\\{A_{n}\\}_{n=1}^{\infty}$ be a sequence of operators such that
$\sup_{n}\|A_{n}\|_{\infty}<1$. Then, we have a pinching

${\cal P}(A)\simeq\bigoplus_{n=1}^{\infty}A_{n}.$

We need two lemmas. The first one is Theorem 5.6.1 for a single strict
contraction:

######  Lemma 5.6.2.

Let $A$ be an operator with $W_{e}(A)\supset{\cal D}$ and let $X$ be a strict
contraction. Then there exists a projection $E$ such that $A_{E}=X$.

The second Lemma is a refined version of the first one:

######  Lemma 5.6.3.

Let $a\geq 1$ and $1>\rho>0$ be two constants. Let $h$ be a norm one vector,
let $X$ be a strict contraction with $\|X\|_{\infty}<\rho$ and let $B$ be an
operator with $\|B\|_{\infty}\leq a$ and $W_{e}(B)\supset{\cal D}$. Then,
there exist a number $\varepsilon>0$, only depending on $\rho$ and $a$, and a
projection $E$ such that: (i) $\dim E=\infty$ and $B_{E}=X$, (ii) $\dim
E^{\perp}=\infty$, $W_{e}(B_{E^{\perp}})\supset{\cal D}$ and
$\|Eh\|\geq\varepsilon$.

Proof of Theorem 5.6.1. The proof is organized in five steps:

Step 1. Some preliminaries are given.

Step 2. Proof of Lemma 5.6.2 in the special case when $X$ is normal,
diagonalizable.

Step 3. Proof of Lemma 5.6.2 in the general case.

Step 4. Proof of Lemma 5.6.3.

Step 5. Conclusion.

1\. Preliminaries

We shall use a sequence $\\{V_{k}\\}_{k\geq 1}$ of orthogonal matrices acting
on spaces of dimensions $2^{k}$. This sequence is built up by induction:

$V_{1}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\ -1&1\end{pmatrix}\quad{\rm
then}\quad V_{k}=\frac{1}{\sqrt{2}}\begin{pmatrix}V_{k-1}&V_{k-1}\\\
-V_{k-1}&V_{k-1}\end{pmatrix}\quad{\rm for}\ k\geq 2.$

Given a Hilbert space ${\cal G}$ and a decomposition

${\cal G}=\bigoplus_{j=1}^{2^{k}}{\cal H}_{j}\quad{\rm with}\ {\cal
H}_{1}=\dots={\cal H}_{2^{k}}={\cal H},$

we may consider the unitary (orthogonal) operator on ${\cal G}$ :
$W_{k}=V_{k}\bigotimes I$, where $I$ denotes the identity on ${\cal H}$,

Now, let $B:{\cal G}\rightarrow{\cal G}$ be an operator which, with respect to
the above decomposition of ${\cal G}$, has a block diagonal matrix

$B=\begin{pmatrix}B_{1}&\ &\ \\\ \ &\ddots&\ \\\ \ &\
&B_{2^{k}}\end{pmatrix}.$

We observe that the block matrix representation of $W_{k}BW_{k}^{*}$ has its
diagonal entries all equal to

$\frac{1}{2^{k}}\left(B_{1}+\dots B_{2^{k}}\right).$

So, the orthogonal operators $W_{k}$ allow us to pass from a block diagonal
matrix representation to a block matrix representation in which the diagonal
entries are all equal.

2\. Proof of Lemma 5.6.2 when $X$ is normal, diagonalizable.

Let $\\{\lambda_{n}(X)\\}_{n\geq 1}$ be the eigenvalues of $X$ repeated
according to their multiplicities. Since $|\lambda_{n}(X)|<1$ for all $n$ and
$W_{e}(A)\supset{\cal D}$, we may find a norm one vector $e_{1}$ such that
$\langle e_{1},Ae_{1}\rangle=\lambda_{1}(T)$. Let $F_{1}=[{\rm
span}\\{e_{1},Ae_{1},A^{*}e_{1}\\}]^{\perp}$. As $F_{1}$ is of finite
codimension, $W_{e}(A_{F_{1}})\supset{\cal D}$. So, there exists a norm one
vector $e_{2}\in F_{1}$ such that $\langle
e_{2},Ae_{2}\rangle=\lambda_{2}(T)$. Next, we set $F_{2}=[{\rm
span}\\{e_{1},Ae_{1},A^{*}e_{1},e_{2},Ae_{2},A^{*}e_{2}\\}]^{\perp}$, $\dots$.
If we go on like this, we exhibit an orthonormal system $\\{e_{n}\\}_{n\geq
1}$ such that, setting $E={\rm span}\\{e_{n}\\}_{n\geq 1}$, we have $A_{E}=X$.

3\. Proof of Lemma 5.6.2 in the general case.

The contraction $Y=(1/\|X\|_{\infty})X$ can be dilated in a unitary

$U=\begin{pmatrix}Y&-(I-YY^{*})^{1/2}\\\ (I-Y^{*}Y)^{1/2}&Y^{*}\end{pmatrix}$

thus $X$ can be dilated in a normal operator $N=\|X\|_{\infty}U$ with
$\|N\|_{\infty}<\rho$. This permits to restrict to the case when $X$ is a
normal strict contraction. So, let $X$ be a normal operator with
$\|X\|_{\infty}<\rho<1$. We remark with the Berg-Weyl-von Neumann theorem,
that $X$ can be written as

$X=D+K$ $None$

where $D$ is normal diagonalizable, $\|D\|_{\infty}=\|X\|_{\infty}<\rho$, and
$K$ is compact with an arbitrarily small norm. Let $K={\rm Re}K+{\rm iIm}K$ be
the Cartesian decomposition of $K$. We can find an integer $l$, a real
$\alpha$ and a real $\beta$ such that decomposition (1) satisfies:

a) the operators $\alpha D$, $\beta{\rm Re}K$, $\beta{\rm Im}K$ are dominated
in norm by $\rho$,

b) there are positive integers $m$, $n$ with $2^{l}=m+2n$ and

$X=\frac{1}{2^{l}}(m\alpha D+n\beta{\rm Re}K+n\beta{\rm iIm}K).$ $None$

More precisely we can take any $l$ such that
$[2^{l}/(2^{l}-2)].\|X\|_{\infty}<\rho$. Next, assuming
$\|K\|_{\infty}<\rho/2^{l}$, we can take $m=2^{l}-2$, $n=1$,
$\alpha=2^{l}/(2^{l}-2)$ and $\beta=2^{l}$.

Let then $T$ be the diagonal normal operator acting on the space

${\cal G}=\bigoplus_{j=1}^{2^{l}}{\cal H}_{j}\quad{\rm with}\ {\cal
H}_{1}=\dots={\cal H}_{2^{l}}={\cal H},$

and defined by

$T=\left(\bigoplus_{j=1}^{m}D_{j}\right)\bigoplus\left(\bigoplus_{j=m+1}^{m+n}R_{j}\right)\bigoplus\left(\bigoplus_{j=m+n+1}^{2^{l}}S_{j}\right)$

where $D_{j}=\alpha D$, $S_{j}=\beta{\rm Re}K$ and $S_{j}=\beta{\rm iIm}K$. We
note that $\|T\|_{\infty}<\rho<1$ and that the operator $W_{l}TW_{l}^{*}$,
represented in the preceding decomposition of ${\cal G}$, has its diagonal
entries all equal to $X$ by (2). Hence, applying the preceding step to $T$
yields Lemma 5.6.2.

4\. Proof of Lemma 5.6.3.

Let $a\geq 1$ and let $1>\rho>0$ be two constants. We take an arbitrary norm
one vector $h$ and any operator $B$ satisfying to the assumptions of Lemma
5.6.3. We can show, using the same reasoning as that applied in the above Step
2, that we have an orthonormal system $\\{f_{n}\\}_{n\geq 0}$, with $f_{0}=h$,
such that:

a) $\langle f_{2j},Bf_{2j}\rangle=0$ for all $j\geq 1$.

b) $\\{\langle f_{2j+1},Bf_{2j+1}\rangle\\}_{j\geq 0}$ is a dense sequence in
${\cal D}$.

c) If $F={\rm span}\\{f_{j}\\}_{j\geq 0}$, then $B_{F}$ is the normal operator

$\sum_{j\geq 0}\langle f_{j},Bf_{j}\rangle f_{j}\otimes f_{j}.$

Setting $F_{0}={\rm span}\\{f_{2j}\\}_{j\geq 0}$ and $F^{\prime}_{0}={\rm
span}\\{f_{2j+1}\\}_{j\geq 0}$, we then have:

a) With respect to the decomposition $F=F_{0}\bigoplus F^{\prime}_{0}$,
$B_{F}$ can be written

$B_{F}=\begin{pmatrix}B_{F_{0}}&0\\\ 0&B_{F^{\prime}_{0}}\end{pmatrix}.$

b) $W_{e}(B_{F^{\prime}_{0}})\supset{\cal D}$ and $h\in F_{0}$.

We can then write a decomposition of $F^{\prime}_{0}$,
$F^{\prime}_{0}=\bigoplus_{j=1}^{\infty}F_{j}$ where for each index $j$,
$F_{j}$ commutes with $B_{F}$ and $W_{e}(B_{F_{j}})\supset{\cal D}$; so that
the decomposition $F=\bigoplus_{j=0}^{\infty}F_{j}$ yields a representation of
$B_{F}$ as a block diagonal matrix,

$B_{F}=\bigoplus_{j=0}^{\infty}B_{F_{j}}.$

Since $W_{e}(B_{F_{j}})\supset{\cal D}$ when $j\geq 1$, the same reasoning as
in Step 3 entails that for any sequence $\\{X_{j}\\}_{j\geq 0}$ of strict
contractions we have decompositions $(\dagger)$ $F_{j}=G_{j}\bigoplus
G^{\prime}_{j}$ allowing us to write, for $j\geq 1$,

$B_{F_{j}}=\begin{pmatrix}X_{j}&*\\\ \ast&*\end{pmatrix}.$

Since $\|X\|_{\infty}<\rho<1$ and $\|B\|_{\infty}\leq a$, we can find an
integer $l$ only depending on $\rho$ and $a$, as well as strict contractions
$X_{1},\dots,X_{2^{l}}$, such that

$X=\frac{1}{2^{l}}\left(B_{F_{0}}+\sum_{j=1}^{2^{l}-1}X_{j}\right).$ $None$

Considering decompositions $(\dagger)$ adapted to these $X_{j}$, we set

$G=F_{0}\bigoplus\left(\bigoplus_{j=1}^{2^{l}-1}G_{j}\right).$

With respect to this decomposition,

$B_{G}=\begin{pmatrix}B_{F_{0}}&\ &\ &\ \\\ \ &X_{1}&\ &\ \\\ \ &\ &\ddots&\
\\\ \ &\ &\ &X_{2^{l}-1}\end{pmatrix}.$

Then we deduce from (3) that the block matrix $W_{l}B_{G}W_{l}^{*}$ has its
diagonal entries all equal to $X$. Summary: $h\in G$ and there exists a
decomposition $G=\bigoplus_{j=1}^{2^{l}}E_{j}$, in which $l$ depends only on
$\rho$ and $a$, such that $B_{E_{j}}=X$ for each $j$. Thus we have an integer
$j_{0}$ such that, setting $E_{j_{0}}=E$, we have

$B_{E}=X\quad{\rm and}\quad\|Eh\|\geq\frac{1}{\sqrt{2^{l}}}.$

Taking $\varepsilon=1/{\sqrt{2^{l}}}$ ends the proof of Lemma 5.6.3.

4\. Conclusion.

Fix a dense sequence $\\{h_{n}\\}$ in the unit sphere of ${\cal H}$ and set
$a=\|A\|_{\infty}$. We claim that the statement (i) and (ii) of Lemma 5.6.3
ensure that there exists a sequence of mutually orthogonal projections
$\\{E_{j}\\}$ such that, setting $F_{n}=\sum_{j\leq n}E_{j}$, we have for all
integers $n$: ($*$) $A_{n}=A_{E_{n}}$ and
$W_{e}(A_{F_{n}^{\perp}})\supset{\cal D}$ (so $\dim F_{n}^{\perp}=\infty$),
($**$) $\|F_{n}h_{n}\|\geq\varepsilon$. In Lemma 5.6.3, set
$a=\|A\|_{\infty}$. Replacing $B$ by $A$, Lemma 5.6.3 proves ($*$) and ($**$)
for $n=1$. Suppose this holds for an $N\geq 1$. Let $\nu(N)\geq N+1$ be the
first integer for which $F_{N}h_{\nu(N)}\neq 0$. Note that
$\|A_{F_{N}^{\perp}}\|_{\infty}\leq\|A\|_{\infty}$. We apply Lemma 5.6.3 to
$B=A_{F_{N}^{\perp}}$, $X=A_{N+1}$ and
$h=F_{N}h_{\nu(N)}/\|F_{N}h_{\nu(N)}\|$. We then deduce that $(*)$ and $(**)$
are still valid for $N+1$. Therefore $(*)$ and $(**)$ hold for all $n$.
Denseness of $\\{h_{n}\\}$ and $(**)$ show that $F_{n}$ strongly increases to
the identity $I$ so that $\sum_{j=1}^{\infty}E_{j}=I$ as required. $\Box$

#####  5.6.3 Müller-Tomilov’s theorem

Define the diagonal set $\Delta(A)$ of an operator $A$ as the scalars
$\lambda\in\mathbb{C}$ such that

$\lambda=\langle e_{n},Ae_{n}\rangle,\ n=1,2,\ldots$

for some orthonormal basis $\\{e_{n}\\}_{n=1}^{\infty}$. One has

${\mathrm{int}}\,W_{e}(A)\subset\Delta(A)\subset W_{e}(A)$

The set $\Delta(A)$ is an analytic set, I do not know examples where
$\Delta(A)$ is not Borel. The boundary of $\Delta(A)$ might be quite
complicated. In 2003 [21], I asked wether $\Delta(A)$ is always convex. Müller
and Tomilov give a positive answer in the recent paper [82].

####  5.7 References of Chapter 5

  * [1

] C.A. Akemann and J. Anderson. Lyapunov theorems for operator algebras. Mem.
Amer. Math. Soc., 94 no 458, 1991.

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] J.H. Anderson and J.G. Stampfli, Commutators and compressions, Israel J.
Math.  10 (1971), 433–441.

  * [5

] T. Ando, Majorization, doubly stochastic matrices, and comparison of
eigenvalues, Linear Algebra Appl.  118 (1989), 163-248.

  * [21

] J.-C. Bourin, Compressions and pinchings, J. Operator Theory 50 (2003), no.
2, 211-220.

  * [33

] J.-C. Bourin and E.-Y. Lee, Sums of Murray-von Neumann equivalent positive
operators, C. R. Math. Acad. Sci. Paris 351 (2013), no. 19-20, 761-764.

  * [34

] J.-C. Bourin and E.-Y. Lee, Sums of unitarily equivalent positive operators,
C. R. Math. Acad. Sci. Paris 352 (2014), no. 5, 435–439.

  * [37

] J.-C. Bourin and E.-Y. Lee, Pinchings and positive linear maps, J. Funct.
Anal. 270 (2016), no. 1, 359–374.

  * [58

] C. Dragan and V. Kaftal, Sums of equivalent sequences of positive operators
in von Neumann factors, preprint, arXiv:1504.03193.

  * [60

] P.R. Halmos, Numerical ranges and normal dilations, Acta Sci. Math. (Szeged)
25 (1964) 1–5.

  * [71

] R. Kadison, The Pythagorean theorem. II. The infinite discrete case, Proc.
Natl. Acad. Sci.  USA 99 (2002), no. 8, 5217-5222.

  * [72

] R. Kadison and I. Singer, Extensions of pure states, Amer. J. Math. (1959),
383-400.

  * [73

] M. Kennedy and P. Skoufranis, The Schur-Horn Problem for Normal Operators,
Proc. London Math. Soc., in press, arXiv:1501.06457.

  * [78

] J. Loreaux and G. Weiss, Diagonality and idempotents with applications to
problems in operator theory and frame theory, J. Operator Theory, in press,
arXiv:1410.7441.

  * [82

] V. Müller, Y. Tomilov, In search of convexity: diagonals and numerical
ranges, Bull. London Math soc., 53 (2021), no. 4, 1016–1029.

  * [85

] C. Pearcy and D. Topping, Sums of small numbers of idempotents, Michigan J.
Math.  14 (1967) 453–465.

  * [87

] M. Ravichandran, The Schur-Horn theorem in von Neumann algebras, preprint,
arXiv:1209.0909.

  * [93

] P.Y. Wu, Additive combination of special operators, Banach Center Publ. 30
(1994), 337-361.

###  Chapter 6 Partial trace

Decomposition and partial trace of positive matrices with Hermitian blocks
[32]

Abstract. Let $H=[A_{s,t}]$ be a positive definite matrix written in
$\beta\times\beta$ Hermitian blocks and let
$\Delta=A_{1,1}+\cdots+A_{\beta,\beta}$ be its partial trace. Assume that
$\beta=2^{p}$ for some $p\in\mathbb{N}$. Then, up to a direct sum operation,
$H$ is the average of $\beta$ matrices isometrically congruent to $\Delta$. A
few corollaries are given, related to important inequalities in quantum
information theory such as the Nielsen-Kempe separability criterion.

Keywords: Positive definite matrices, norm inequalities, partial trace,
separable state.

AMS subjects classification 2010: 15A60, 47A30, 15A42.

####  6.1 Introduction and a key lemma

Positive semi-definite matrices partitioned in two by two blocks occur as an
efficient tool in matrix analysis, sometimes a magic tool ! $-$ according to
Bhatia’s famous book [13]. These partitions allow to derive a lot of important
inequalities and those with Hermitian blocks shed much light on the geometric
and harmonic matrix means. Partitions into a larger number of blocks are
naturally involved with tensor products, in the theory of positive linear maps
and in their application in quantum physics.

This article deals with positive matrices partitioned in Hermitian blocks. By
using unitary or isometry congruences, we will improve some nice
majorisations, or norm estimates, first obtained in the field of quantum
information theory.

For partitioned positive matrices, the diagonal blocks play a quite special
role. This is apparent in a rather striking decomposition due to the authors
[31].

######  Lemma 6.1.1.

For every matrix in $\mathbb{M}_{n+m}^{+}$ written in blocks, we have a
decomposition

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

for some unitaries $U,\,V\in\mathbb{M}_{n+m}$.

This lemma leads to study partitions via unitary congruences. It is the key of
the subsequent results. A proof and several consequences can be found in [31]
and [28]. Of course, $\mathbb{M}_{n}$ is the algebra of $n\times n$ matrices
with real or complex entries, and $\mathbb{M}_{n}^{+}$ is the positive part.
That is, $\mathbb{M}_{n}$ may stand either for $\mathbb{M}_{n}(\mathbb{R})$,
the matrices with real entries, or for $\mathbb{M}_{n}(\mathbb{C})$, those
with complex entries. The situation is different in the next statement, where
complex entries seem unavoidable.

######  Theorem 6.1.2.

Given any matrix in $\mathbb{M}_{2n}^{+}(\mathbb{C})$ written in blocks in
$\mathbb{M}_{n}(\mathbb{C})$ with Hermitian off-diagonal blocks, we have

$\begin{bmatrix}A&X\\\
X&B\end{bmatrix}=\frac{1}{2}\left\\{U(A+B)U^{*}+V(A+B)V^{*}\right\\}$

for some isometries $U,V\in\mathbb{M}_{2n,n}(\mathbb{C})$.

Here $\mathbb{M}_{p,q}(\mathbb{C})$ denote the space of $p$ rows and $q$
columns matrices with complex entries, and $V\in\mathbb{M}_{p,q}(\mathbb{C})$
is an isometry if $p\geq q$ and $V^{*}V=I_{q}$. Even for a matrix in
$\mathbb{M}_{2n}^{+}(\mathbb{R})$, it seems essential to use isometries with
complex entries ! The result, due to Lin and the authors, is based on Lemma
6.1.1, a proof is in [46] and implicitly in [45].

There is no evidence whether a positive block-matrix $H$ in
$\mathbb{M}^{+}_{3n}$,

$H=\begin{bmatrix}A&X&Y\\\ X&B&Z\\\ Y&Z&C\end{bmatrix}$

with Hermitian off-diagonal blocks $X,Y,Z$, could be decomposed as

$H=\frac{1}{3}\left\\{U\Delta U^{*}+V\Delta V^{*}+W\Delta W^{*}\right\\}$

where $\Delta=A+B+C$ and $U,V,W$ are isometries. In fact, this would be
surprising. However, a quite nice decomposition is possible by considering
direct sum copies: this provides a substitute to Theorem 6.1.2 for partitions
into an arbitrary number of blocks; it is the main result of this article.

These decompositions entail some nice inequalities. Lemma 6.1.1 yields a
simple estimate for all symmetric (or unitarily invariant) norms,

$\left\|\begin{bmatrix}A&X\\\ X^{*}&B\end{bmatrix}\right\|\leq\|A\|+\|B\|.$
(6.1.1)

Recall that a symmetric norm on $\mathbb{M}_{m}$ satisfies
$\|A\|=\|UA\|=\|AU\|$ for all $A\in\mathbb{M}_{m}$ and all unitaries
$U\in\mathbb{M}_{m}$. This obviously induces a symmetric norm on
$\mathbb{M}_{n}$, $1\leq n\leq m$. The most familiar symmetric norms are the
Schatten $p$-norms, $1\leq p<\infty$,

$\|A\|_{p}=\\{{\mathrm{Tr\,}}(A^{*}A)^{p/2}\\}^{1/p},$ (6.1.2)

and, with $p\to\infty$, the operator norm. In general, the sum of the norms
$\|A\|+\|B\|$ can not be replaced in (6.1.1) by the norm of the sum $\|A+B\|$.
However, Theorem 6.1.2 implies the following remarkable corollary.

######  Corollary 6.1.3.

Given any matrix in $\mathbb{M}_{2n}^{+}$ written in blocks in
$\mathbb{M}_{n}$ with Hermitian off-diagonal blocks, we have

$\left\|\begin{bmatrix}A&X\\\ X&B\end{bmatrix}\right\|\leq\|A+B\|$

for all symmetric norms.

This is the simplest case of Hiroshima’s theorem, discussed in the next
section. There are some positive matrices in $\mathbb{M}_{6}$ partitioned in
blocks in $\mathbb{M}_{3}$, with normal off-diagonal blocks $X$, $X^{*}$, such
that

$\left\|\begin{pmatrix}A&X\\\
X^{*}&B\end{pmatrix}\right\|_{\infty}>\|A+B\|_{\infty}.$

Hence the assumptions are rather optimal.

In Section 2, we state our decomposition and derive several inequalities, most
of them related to Hiroshima’s theorem. The proof of the decomposition is
given in Section 3. A discussion of previous results for small partitions and
some remarks related to quantum information are given in the last section.

####  6.2 Direct sum and partial trace

A typical example of positive matrices written in blocks are formed by tensor
products. Indeed, the tensor product $A\otimes B$ of $A\in\mathbb{M}_{\beta}$
with $B\in\mathbb{M}_{n}$ can be identified with an element of
$\mathbb{M}_{\beta}(\mathbb{M}_{n})=\mathbb{M}_{\beta n}$. Starting with
positive matrices in $\mathbb{M}_{\beta}^{+}$ and $\mathbb{M}_{n}^{+}$ we then
get a matrix in $\mathbb{M}_{\beta n}^{+}$ partitioned in blocks in
$\mathbb{M}_{n}$. In quantum physics, sums of tensor products of positive
semi-definite (with trace one) occur as so-called separable states. In this
setting of tensor products, the sum of the diagonal block is called the
partial trace (with respect to $\mathbb{M}_{\beta}$). We will use this
terminology.

######  Theorem 6.2.1.

Let $H=[A_{s,t}]\in\mathbb{M}_{\beta n}^{+}$ be written in $\beta\times\beta$
Hermitian blocks in $\mathbb{M}_{n}$ and let
$\Delta=\sum_{s=1}^{\beta}{A_{s,s}}$ be its partial trace. If $\beta$ is
dyadic, then, with $m=2^{\beta}$, we have

$\oplus^{m}H=\frac{1}{\beta}\sum_{k=1}^{\beta}V_{k}\left(\oplus^{m}\Delta\right)V_{k}^{*}$

where $\\{V_{k}\\}_{k=1}^{\beta}$ is a family of isometries in
$\mathbb{M}_{m\beta n,mn}$.

Here the spaces $\mathbb{M}_{n}$ and $\mathbb{M}_{p,q}$ denote either the real
or complex spaces of matrices. By a dyadic number $\beta$, we mean
$\beta=2^{p}$ for some $p\in\mathbb{N}$.

This theorem has strong links with quantum information theory (QIT) as
detailed in Section 4. Researchers in QIT may like to restate the theorem by
replacing direct sums with tensors products, $\oplus^{m}H\rightarrow
I_{m}\otimes H$ and $\oplus^{m}\Delta\rightarrow I_{m}\otimes\Delta$.
Tensoring in identity means that an operator on a Hilbert space
${\mathcal{H}}$ is lifted to an operator acting on
${\mathcal{F}}\otimes{\mathcal{H}}$ where ${\mathcal{F}}$ is an auxiliary
Hilbert space, an ancilla space in the QIT terminology.

#####  6.2.1 Around Hiroshima’s theorem

A straightforward application of Theorem 6.2.1 is the following beautiful
result first proved by Hiroshima in 2003 (see Section 4 for the complete form
of Hiroshima’s theorem and is relevance in quantum physics).

######  Corollary 6.2.2.

Let $H=[A_{s,t}]\in\mathbb{M}_{\alpha n}^{+}$ be written in
$\alpha\times\alpha$ Hermitian blocks in $\mathbb{M}_{n}$ and let
$\Delta=\sum_{s=1}^{\alpha}{A_{s,s}}$ be its partial trace. Then, we have

$\left\|H\right\|\leq\left\|\Delta\right\|$

for all symmetric norms.

###### Proof.

By completing $H$ with some zero rows and columns, we may assume that
$\alpha=\beta$ is dyadic. Theorem 6.2.1 then implies, with $m=2^{\beta}$,

$\|\oplus^{m}H\|\leq\|\oplus^{m}\Delta\|$

for all symmetric norms, which is equivalent to the claim of the corollary. ∎

By the Ky Fan principle, Corollary 6.2.2 is equivalent to the majorisation
relation

$\sum_{i=1}^{j}\lambda_{i}(H)\leq\sum_{i=1}^{j}\lambda_{i}(\Delta)$

for all $j=1,\ldots,\alpha n$. (we set $\lambda_{j}(A)=0$ when
$A\in\mathbb{M}_{d}^{+}$ and $j>d$).

Theorem 6.2.1 says much more than this majorisation. For instance, we may
completes these eigenvalue relations with the following ones.

######  Corollary 6.2.3.

Let $H=[A_{s,t}]\in\mathbb{M}_{\alpha n}^{+}$ be written in
$\alpha\times\alpha$ Hermitian blocks in $\mathbb{M}_{n}$ and let
$\Delta=\sum_{s=1}^{\alpha}{A_{s,s}}$ be its partial trace. Then, we have

$\lambda_{1+\beta k}(H)\leq\lambda_{1+k}(\Delta)$

for all $k=0,\ldots,n-1$, where $\beta$ is the smallest dyadic number such
that $\alpha\leq\beta$.

###### Proof.

By completing $H\in\mathbb{M}_{\alpha n}^{+}$ with some zero blocks, we may
assume that $H\in\mathbb{M}_{\beta n}^{+}$. Theorem 6.2.1 then yields the
decomposition

$\oplus^{m}H=\frac{1}{\beta}\sum_{k=1}^{\beta}V_{k}\left(\oplus^{m}\Delta\right)V_{k}^{*}$

where $\\{V_{k}\\}_{k=1}^{\beta}$ is a family of isometries in
$\mathbb{M}_{m\beta n,mn}$ and $m=2^{\beta}$. We recall a simple fact, Weyl’s
theorem: if $Y,Z\in\mathbb{M}_{d}$ are Hermitian, then

$\lambda_{r+s+1}(Y+Z)\leq\lambda_{r+1}(Y)+\lambda_{s+1}(Z)$

for all nonnegative integers $r,s$ such that $r+s\leq d-1$. When $Y,Z$ are
positive, this still holds for all nonnegative integers $r,s$ with our
convention ($\lambda_{j}(A)=0$ when $A\in\mathbb{M}_{d}^{+}$ and $j>d$). From
the previous decomposition we thus infer

$\lambda_{1+\beta
k}\left(\oplus^{m}H\right)\leq\lambda_{1+k}\left(\oplus^{m}\Delta\right)$
(6.2.1)

for all $k=0,1,\ldots$. Then, observe that for all $A\in\mathbb{M}_{d}^{+}$
and all $j=0,1,\ldots$,

$\lambda_{1+j}\left(\oplus^{m}A\right)=\lambda_{\langle(1+j)/m\rangle}(A)$
(6.2.2)

where $\langle u\rangle$ stands for the smallest integer greater than or equal
to $u$. Combining (6.2.1) and (6.2.2) we get

$\lambda_{\langle(1+\beta
j)/m\rangle}\left(H\right)\leq\lambda_{\langle(1+j)/m\rangle}\left(\Delta\right)$

for all $j=0,1,\ldots$. Taking $j=km$, $k=0,1\ldots$ completes the proof. ∎

The above proof actually shows more eigenvalue inequalities.

######  Corollary 6.2.4.

Let $H=[A_{s,t}]\in\mathbb{M}_{\alpha n}^{+}$ be written in
$\alpha\times\alpha$ Hermitian blocks in $\mathbb{M}_{n}$ and let
$\Delta=\sum_{s=1}^{\alpha}{A_{s,s}}$ be its partial trace. Then, we have

$\lambda_{1+\beta
k}(S)\leq\frac{1}{\beta}\left\\{\lambda_{1+k_{1}}\left(\Delta\right)+\cdots+\lambda_{1+k_{\beta}}\left(\Delta\right)\right\\}$

where $k_{1}+\cdots+k_{\beta}=\beta k$ and $\beta$ is the smallest dyadic
number such that $\alpha\leq\beta$.

Corollary 6.2.2 implies the following rearrangement inequality.

######  Corollary 6.2.5.

Let $\\{S_{i}\\}_{i=1}^{\alpha}$ be a commuting family of Hermitian operators
in $\mathbb{M}_{n}$ and let $T\in\mathbb{M}_{n}^{+}$. Then,

$\left\|\sum_{i=1}^{\alpha}S_{i}T^{2}S_{i}\right\|\leq\left\|\sum_{i=1}^{\alpha}TS^{2}_{i}T\right\|$

for all symmetric norms.

###### Proof.

Define a matrix $Z\in\mathbb{M}_{\alpha n}$ by

$Z=XX^{*}=\begin{bmatrix}TS_{1}\\\ \vdots\\\
TS_{\alpha}\end{bmatrix}\begin{bmatrix}S_{1}T&\cdots&S_{\alpha}T\end{bmatrix}.$

Hence $Z=[TS_{i}S_{j}T]$ is positive and partitioned in Hermitian blocks in
$\mathbb{M}_{n}$, with diagonal blocks $TS_{i}^{2}T$, $1\leq i\leq\alpha$.
Thus, for all symmetric norms,

$\|Z\|\leq\left\|\sum_{i=1}^{\alpha}TS_{i}^{2}T\right\|$

Since $XX^{*}$ and $X^{*}X$ have same symmetric norms for any rectangular
matrix $X$, we infer

$\left\|\sum_{i=1}^{\alpha}S_{i}T^{2}S_{i}\right\|\leq\left\|\sum_{i=1}^{\alpha}TS_{i}^{2}T\right\|$

as claimed. ∎

From Corollary 6.2.3 we similarly get the next one.

######  Corollary 6.2.6.

Let $\\{S_{i}\\}_{i=1}^{\alpha}$ be a commuting family of Hermitian operators
in $\mathbb{M}_{n}$ and let $T\in\mathbb{M}_{n}^{+}$. Then,

$\lambda_{1+\beta
k}\left(\sum_{i=1}^{\alpha}S_{i}T^{2}S_{i}\right)\leq\lambda_{1+k}\left(\sum_{i=1}^{\alpha}TS^{2}_{i}T\right)$

for all $k=0,\ldots,n-1$, where $\beta$ is the smallest dyadic number such
that $\alpha\leq\beta$.

#####  6.2.2 Around Rotfel’d inequality

Given two Hermitian matrices $A$, $B$ in $\mathbb{M}_{n}$ and a concave
function $f(t)$ defined on the real line,

${\mathrm{Tr\,}}f\left(\frac{A+B}{2}\right)\geq{\mathrm{Tr\,}}\frac{f(A)+f(B)}{2}$
(6.2.3)

and, if further $f(0)\geq 0$ and both $A$ and $B$ are positive semi-definite,

${\mathrm{Tr\,}}f(A+B)\leq{\mathrm{Tr\,}}f(A)+{\mathrm{Tr\,}}f(B).$ (6.2.4)

The first inequality goes back to von-Neumann in the 1920’s, the second is
more subtle and has been proved only in 1969 by Rotfel’d [88]. These trace
inequalities are matrix versions of obvious scalar inequalities. Theorem 6.2.1
yields a refinement of the Rotfel’d inequality for families of positive
operators $\\{A_{i}\\}_{i=1}^{\alpha}$ by considering these operators as the
diagonal blocks of a partitioned matrix $H$ as follows.

######  Corollary 6.2.7.

Let $H=[A_{s,t}]\in\mathbb{M}_{\alpha n}^{+}$ be written in
$\alpha\times\alpha$ Hermitian blocks in $\mathbb{M}_{n}$. Then, we have

${\mathrm{Tr}}\,f\left(\sum_{s=1}^{\alpha}A_{s,s}\right)\leq{\mathrm{Tr}}\,f(H)\leq\sum_{s=1}^{\alpha}{\mathrm{Tr}}\,f(A_{s,s})$

for all concave functions $f(t)$ on $\mathbb{R}^{+}$ such that $f(0)\geq 0$.

###### Proof.

Note that if these inequalities hold for a non-negative concave function
$f(t)$ with $f(0)=0$, then they also hold for the function $f(t)+c$ for any
constant $c>0$. Therefore it suffice to consider concave functions vanishing
at the origine. This assumption entails that

$f(VAV^{*})=Vf(A)V^{*}$ (6.2.5)

for all $A\in\mathbb{M}_{n}^{+}$ and all isometries $V\in\mathbb{M}_{m,n}$. By
Lemma 6.1.1 we have a decomposition

$H=\sum_{s=1}^{\alpha}V_{s}A_{s,s}V_{s}^{*}$

for some isometries $V_{s}\in\mathbb{M}_{\alpha n,n}$. Inequality (6.2.4) then
yields

${\mathrm{Tr}}\,f(H)\leq\sum_{s=1}^{\alpha}{\mathrm{Tr}}\,f(V_{s}A_{s,s}V_{s}^{*})$

and using (6.2.5) establishes the second inequality. To prove the first
inequality, we use Theorem 6.2.1. By completing $H$ with some zero blocks (we
still suppose $f(0)=0$) we may assume that $\alpha=\beta$ is dyadic. Thus we
have a decomposition

$\oplus^{m}H=\frac{1}{\beta}\sum_{k=1}^{\beta}V_{k}\left(\oplus^{m}\Delta\right)V_{k}^{*}$

where $\\{V_{k}\\}_{k=1}^{\beta}$ is a family of isometries in
$\mathbb{M}_{m\beta n,mn}$ and $m=2^{\beta}$. Inequality (6.2.3) then gives

${\mathrm{Tr}}\,f\left(\oplus^{m}H\right)\geq\frac{1}{\beta}\sum_{k=1}^{\beta}{\mathrm{Tr}}\,f\left(V_{k}\left(\oplus^{m}\Delta\right)V_{k}^{*}\right)$

and using (6.2.5) we obtain

${\mathrm{Tr}}\ f\left(\oplus^{m}H\right)\geq{\mathrm{Tr}}\
f\left(\oplus^{m}\Delta\right).$

The proof is completed by dividing both sides by $m$. ∎

######  Remark 6.2.8.

The first inequality of Corollary 6.2.7 is actually equivalent to Corollary
6.2.2 by a well-known majorisation principle for convex/concave functions. The
above proof does not require this principle. The simplest case of Corollary
6.2.7 is the double inequality

$f(a_{1}+\cdots+a_{n})\leq{\mathrm{Tr}}\,f(A)\leq f(a_{1})+\cdots+f(a_{n})$

for all $A\in\mathbb{M}_{n}^{+}$ with diagonal entries $a_{1},\ldots,a_{n}$.

######  Remark 6.2.9.

A special case of Corollary 6.2.7 refines a well-known determinantal
inequality. Taking as a concave function on $\mathbb{R}^{+}$,
$f(t)=\log(1+t)$, we obtain: Let $A,B\in\mathbb{M}_{n}^{+}$. Then, for any
Hermitian $X\in\mathbb{M}_{n}$ such that

$H=\begin{bmatrix}A&X\\\ X&B\end{bmatrix}$

is positive semi-definite, we have

$\det(I+A+B)\leq\det(I+H)\leq\det(I+A)\det(I+B).$

This was noted in [46].

####  6.3 Proof of Theorem 2.1

A Clifford algebra ${\mathcal{C}}_{\beta}$ is the associative real algebra
generated by $\beta$ elements $q_{1},\ldots,q_{\beta}$ satisfying the
canonical anticommutation relations $q_{i}^{2}=1$ and

$q_{i}q_{j}+q_{j}q_{i}=0$

for $i\neq j$. This structure was introduced by Clifford in [53]. It turned
out to be of great importance in quantum theory and operator algebras, for
instance see the survey [57]. From the relation

$\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}+\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}\begin{pmatrix}0&1\\\
1&0\end{pmatrix}=0$

we infer a representation of ${\mathcal{C}}_{\beta}$ as a a real subalgebra of
$M_{2^{\beta}}=\otimes^{\beta}\mathbb{M}_{2}$ by mapping the generators
$q_{j}\mapsto Q_{j}$, $1\leq j\leq\beta$, where

$Q_{j}=\left\\{\otimes^{j-1}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\right\\}\otimes\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\otimes\left\\{\otimes^{\beta-j}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\right\\}.$ (6.3.1)

We use these matrices in the following proof of Theorem 6.2.1.

###### Proof.

First, replace the positive block matrix $H=[A_{s,t}]$ where $1\leq
s,t,\leq\beta$ and all blocks are Hermitian by a bigger one in which each
block in counted $2^{\beta}$ times :

$G=[G_{s,t}]:=\left[I_{2^{\beta}}\otimes
A_{s,t}\right]=\left[\oplus^{2^{\beta}}A_{s,t}\right]$

where $I_{r}$ stands for the identity of $\mathbb{M}_{r}$. Thus
$G\in\mathbb{M}_{\beta 2^{\beta}n}$ is written in $\beta$-by-$\beta$ blocks in
$\mathbb{M}_{2^{\beta}n}$. Then perform a unitary congruence with the unitary
$W\in\mathbb{M}_{\beta 2^{\beta}n}$ defined as

$W=\bigoplus_{j=1}^{\beta}\left\\{Q_{j}\otimes I_{n}\right\\}$ (6.3.2)

where $Q_{j}$ is given by (6.3.1), $1\leq j\leq\beta$. Thanks to the
anticommutation relation for each pair of summands in (6.3.2),

$\left\\{Q_{j}\otimes I_{n}\right\\}\left\\{Q_{l}\otimes
I_{n}\right\\}+\left\\{Q_{l}\otimes I_{n}\right\\}\left\\{Q_{j}\otimes
I_{n}\right\\}=0,\quad j\neq l,$

the block matrix (with $W=W^{*}$)

$\Omega:=WGW^{*}=[\Omega_{s,t}]$ (6.3.3)

satisfies the following : For $1\leq s<t\leq\beta$,

$\Omega_{s,t}=-\Omega_{t,s}.$ (6.3.4)

Next, consider the reflexion matrix

$J_{1}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\ 1&-1\end{pmatrix}$

and define inductively for all integers $p>1$, a reflexion

$J_{p}=\frac{1}{\sqrt{2}}\begin{pmatrix}J_{p-1}&J_{p-1}\\\
J_{p-1}&-J_{p-1}\end{pmatrix},$

that is $J_{p}=\otimes^{p}J_{1}$. Observe, that given any matrix
$S\in\mathbb{M}_{2^{p}}$, $S=[s_{i,j}]$, such that $s_{i,j}=-s_{i,j}$ for all
$i\neq j$, the matrix

$T=J_{p}SJ_{p}^{*}$

has its diagonal entries $t_{j,j}$ all equal to the normalized trace
$2^{-p}{\mathrm{Tr\,}}S$. Indeed, letting $J_{p}=[z_{i,j}]$,

$\displaystyle t_{j,j}$
$\displaystyle=\sum_{k}z_{j,k}\left(\sum_{l}s_{k,l}z_{l,j}\right)$
$\displaystyle=\sum_{k,l}z_{j,k}s_{k,l}z_{l,j}$
$\displaystyle=\sum_{k}z_{j,k}s_{k,k}z_{k,j}+\sum_{k\neq
l}z_{j,k}s_{k,l}z_{l,j}$
$\displaystyle=2^{-p}{\mathrm{Tr\,}}S+\sum_{k<l}\left(s_{k,l}z_{j,k}z_{l,j}+s_{l,k}z_{l,j}z_{k,j}\right)$
$\displaystyle=2^{-p}{\mathrm{Tr\,}}S.$

Now, since we assume that $\beta=2^{p}$ for some integer $p$, we may perform a
unitary congruence to the matrix $\Omega$ in (6.3.3) with the unitary matrix

$R_{p}=J_{p}\otimes I_{2^{\beta}}\otimes I_{n}$

and, making use of (6.3.4) and the above property of $J_{p}$, we note that
$R_{p}\Omega R_{p}^{*}$ has its $\beta$ diagonal blocks $(R_{p}\Omega
R_{p}^{*})_{j,j}$, $1\leq j\leq\beta$, all equal to the matrix
$D\in\mathbb{M}_{2^{\beta}n}$,

$D=\frac{1}{\beta}\sum_{s=1}^{\beta}\left\\{\oplus^{2^{\beta}}A_{s,s}\right\\}.$

Thanks to the decomposition of Lemma 6.1.1 and its obvious extension for
$\beta\times\beta$ partitions, there exist some isometries
$U_{k}\in\mathbb{M}_{\beta 2^{\beta}n,2^{\beta}n}$, $1\leq k\leq\beta$, such
that

$\Omega=\sum_{k=1}^{\beta}U_{k}DU_{k}^{*}.$

Since $\Omega$ is unitarily equivalent to $\oplus^{2^{\beta}}H$, that is
$\Omega=V^{*}(\oplus^{2^{\beta}}H)V$ for some unitary $V\in\mathbb{M}_{\beta
2^{\beta}n,2^{\beta}n}$, we get

$\oplus^{2^{\beta}}H=\sum_{k=1}^{\beta}VU_{k}DU_{k}^{*}V^{*}$

wich is the claim of Theorem 6.2.1 by setting $VU_{k}=:V_{k}$, $1\leq
k\leq\beta$, as $2^{\beta}=m$, and $D=\frac{1}{\beta}\oplus^{m}\Delta$. ∎

####  6.4 Comments

#####  6.4.1 Complex matrices and small partitions

If one uses isometries with complex entries, then, in case of partitions into
a small number of $\beta\times\beta$ blocks, the number $m$ of copies in the
direct sum $\oplus^{m}H$ and $\oplus^{m}\Delta$ can be reduced. For $\beta=2$,
Theorem 6.1.2 shows that it suffices to take $m=1$. For $\beta=3$ or $\beta=4$
the following result holds [46].

######  Theorem 6.4.1.

Let $H=[A_{s,t}]\in\mathbb{M}_{\beta n}^{+}(\mathbb{C})$ be written in
Hermitian blocks in $\mathbb{M}_{n}(\mathbb{C})$ with $\beta\in\\{3,4\\}$ and
let $\Delta=\sum_{s=1}^{\beta}A_{s,s}$ be its partial trace. Then,

$H\oplus
H=\frac{1}{4}\sum_{k=1}^{4}V_{k}\left(\Delta\oplus\Delta\right)V_{k}^{*}$

for some isometries $V_{k}\in\mathbb{M}_{2\beta n,2n}(\mathbb{C})$,
$k=1,2,3,4$.

Likewise for Theorem 6.1.2, we must consider isometries with complex entries,
even for a full matrix $H$ with real entries. The proof makes use of
quaternions and thus confines to $\beta\leq 4$.

#####  6.4.2 Separability criterion

Let ${\mathcal{H}}$ and ${\mathcal{F}}$ be two finite dimensional Hilbert
spaces that may be either real spaces, identified to $\mathbb{R}^{n}$ and
$\mathbb{R}^{m}$, or complex spaces, identified to $\mathbb{C}^{n}$ and
$\mathbb{C}^{m}$. The space of operators on ${\mathcal{H}}$, denoted by
${\mathrm{B}}({\mathcal{H}})$, is identified with the matrix algebra
$\mathbb{M}_{n}$ (with real or complex entries according the nature of
${\mathcal{H}}$). A positive (semi-definite) operator $Z$ on the tensor
product space ${\mathcal{H}}\otimes{\mathcal{F}}$ is said to be separable if
it can be decomposed as a sum of tensor products of positive operators,

$Z=\sum_{j=1}^{k}A_{j}\otimes B_{j}$ (6.4.1)

where $A_{j}$’s are positive operators on ${\mathcal{H}}$ and so $B_{j}$’s are
on ${\mathcal{F}}$ (the positivity assumption on the $A_{j}$’s and $B_{j}$’s
is essential, otherwise 6.4.1 is always possible for any $Z$). It is difficult
in general to determine if a given positive operator in the matrix algebra
$\mathbb{M}_{n}\otimes\mathbb{M}_{m}$ is separable or not, though some
theoretical criteria do exist [70], [51]. The partial trace of $Z$ with
respect to ${\mathcal{H}}$ is the operator acting on ${\mathcal{F}}$,

${\mathrm{Tr}}_{\mathcal{H}}Z=\sum_{j=1}^{k}({\mathrm{Tr}}A_{j})B_{j}.$

These notions have their own mathematical interest and moreover play a
fundamental role in the description of bipartite systems in quantum theory,
see [86, Chapter 10], where the positive operators act on complex spaces and
are usually normalized with trace one and called states. Thus a separable
state is an operator of the type (6.4.1) with ${\mathrm{Tr\,}}Z=1$. The
richness of the mathematical theory of separable operators/states and their
application in quantum physics is apparent in many places in the literature,
for instance in [70] and [2]. Nielsen and Kempe in 2001 proved a majorisation
separability criterion [84]. It can be stated as the following norm
comparison.

######  Theorem 6.4.2.

Let $Z$ be a separable state on the tensor product of two finite dimensional
Hilbert spaces ${\mathcal{H}}$ and ${\mathcal{F}}$. Then, for all symmetric
norms,

$\|Z\|\leq\left\|{\mathrm{Tr}}_{\mathcal{H}}Z\right\|.$

Regarding ${\mathrm{B}}({\mathcal{H}}\otimes{\mathcal{F}})$ as
$\mathbb{M}_{n}(\mathbb{M}_{m})$, an operator
$Z\in{\mathrm{B}}({\mathcal{H}}\otimes{\mathcal{F}})$ is written as a block-
matrix $Z=[Z_{i,j}]$ with $Z_{i,j}\in\mathbb{M}_{m}$, $1\leq i,j\leq n$. The
partial trace of $Z$ with respect to ${\mathcal{H}}$ is then the sum of the
diagonal blocks,

${\mathrm{Tr}}_{\mathcal{H}}Z=\sum_{j=1}^{n}Z_{j,j}.$

This observation makes obvious that Theorem 6.4.2 is a straightforward
consequence of Corollary 6.2.2 whenever the factor ${\mathcal{H}}$ is a real
Hilbert space. Indeed, we then have

$Z=\sum_{j=1}^{k}A_{j}\otimes B_{j}$

where, for each index $j$, $A_{j}\in\mathbb{M}_{n}(\mathbb{R})$ and $B_{j}$ is
Hermitian in $\mathbb{M}_{m}$, so that $A_{j}\otimes B_{j}$ can be regarded as
an element of $\mathbb{M}_{n}(\mathbb{M}_{m})$ formed of Hermitian blocks.

From Corollary 6.2.3, we may complete the majorisation of Theorem 6.4.2, when
a factor is a real space with a few more eigenvalue estimate as stated in the
next corollary.

######  Corollary 6.4.3.

Let $Z$ be a separable positive operator on the tensor product of two finite
dimensional Hilbert space ${\mathcal{H}}\otimes{\mathcal{F}}$ with a real
factor ${\mathcal{H}}$. Then,

$\lambda_{1+\beta
k}(Z)\leq\lambda_{1+k}\left({\mathrm{Tr}}_{\mathcal{H}}Z\right)$

for all $k=0,\ldots,\dim{\mathcal{F}}-1$, where $\beta$ is the smallest dyadic
number such that $\dim{\mathcal{H}}\leq\beta$.

Similarly, from Corollary 6.2.4, we actually have a larger set of eigenvalue
inequalities.

######  Corollary 6.4.4.

Let $Z$ be a separable positive operator on the tensor product of two finite
dimensional Hilbert space ${\mathcal{H}}\otimes{\mathcal{F}}$ with a real
factor ${\mathcal{H}}$. Then,

$\lambda_{1+\beta
k}(Z)\leq\frac{1}{\beta}\left\\{\lambda_{1+k_{1}}\left({\mathrm{Tr}}_{\mathcal{H}}Z\right)+\cdots+\lambda_{1+k_{\beta}}\left({\mathrm{Tr}}_{\mathcal{H}}Z\right)\right\\}$

for all $k=0,\ldots,\dim{\mathcal{F}}-1$, where $k_{1}+\cdots+k_{\beta}=\beta
k$ and $\beta$ is the smallest dyadic number such that
$\dim{\mathcal{H}}\leq\beta$.

Of course, confining to real spaces is a severe restriction. It would be
desirable to obtain similar estimates for usual complex spaces. A related
problem would be to obtain a decomposition like in Theorem 6.2.1 for the class
of partitioned matrices considered in the full form of Hiroshima’s theorem: If
$A=[A_{s,t}]$ is a positive matrix partitioned in $\alpha\times\alpha$ blocks
such that $B=[B_{s,t}]:=[A_{s,t}^{T}]$ is positive too, then the majorisation
of Corollary 6.2.2 holds. Here $X^{T}$ means the transposed matrix. This
statement extends Corollary 6.2.2 and implies Theorem 6.4.2.

####  6.5 Around this article

A companion result of Lemma 6.1.1 is given in [45] :

######  Lemma 6.5.1.

Let $\begin{bmatrix}A&X\\\ X^{*}&B\end{bmatrix}$ be a positive matrix
partitioned into four blocks in $\mathbb{M}_{n}$. Then, for some unitary
$U,V\in\mathbb{M}_{2n}$,

$\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}=U\begin{bmatrix}\frac{A+B}{2}+{\mathrm{Re}}\,X&0\\\
0&0\end{bmatrix}U^{*}+V\begin{bmatrix}0&0\\\
0&\frac{A+B}{2}-{\mathrm{Re}}\,X\end{bmatrix}V^{*}$

.

This lemma is the main tool to obtain original estimates between the full
matrix and its partial trace $A+B$. These estimates involve the geometry of
the numerical range $W(X)$. Here we state two theorems, the next chapter will
develop this topic in more details. The main theorem of [47] reads as follows.

######  Theorem 6.5.2.

Let $\begin{bmatrix}A&X\\\ X^{*}&B\end{bmatrix}$ be a positive matrix
partitioned into four blocks in $\mathbb{M}_{n}$. Suppose that $W(X)$ has the
width $\omega$. Then, for all symmetric norms,

$\left\|\begin{bmatrix}A&X\\\ X^{*}&B\end{bmatrix}\right\|\leq\|A+B+\omega
I\|.$

Here the width of $W(X)$ is the smallest distance between two parallel
straight lines such that the strip between these two lines contains $W(X)$.
Hence the partial trace $A+B$ may be used to give an upper bound for the norms
of the full block-matrix.

A lower bound is given in [41] in which the distance from $0$ to $W(X)$
contributes. We state the main result of [41].

######  Theorem 6.5.3.

Let $\begin{bmatrix}A&X\\\ X^{*}&B\end{bmatrix}$ be a positive matrix
partitioned into four blocks in $\mathbb{M}_{n}$ and let
$d={\mathrm{dist}}(0,W(X))$. Then, for all symmetric norms,

$\left\|\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}\right\|\geq\left\|\left(\frac{A+B}{2}+dI\right)\oplus\left(\frac{A+B}{2}-dI\right)\right\|.$

Several consequences follow. We mention two corollaries

######  Corollary 6.5.4.

For every positive matrix partitioned into four blocks of same size,

${\mathrm{diam}}\,W\left(\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}\right)-{\mathrm{diam}}\,W\left(\frac{A+B}{2}\right)\geq
2d,$

where $d$ is the distance from $0$ to $W(X)$.

######  Corollary 6.5.5.

Let $\begin{bmatrix}A&X\\\ X^{*}&B\end{bmatrix}$ be a positive matrix
partitioned into four blocks in $\mathbb{M}_{n}$ and let
$d={\mathrm{dist}}(0,W(X))$. Then,

$\det\left\\{\left(\frac{A+B}{2}\right)^{2}-d^{2}I\right\\}\geq\det\left(\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}\right).$

Letting $X=0$, we recapture a basic property: the determinant is a log-concave
map on the positive cone of $\mathbb{M}_{n}$. Hence Corollary 6.5.5 refines
this property.

We will see in the next, short chapter several eigenvalue inequalities
associated to Theorem 7.1.1.

####  6.6 References of Chapter 6

  * [13

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

  * [28

] J.-C. Bourin and F. Hiai, Norm and anti-norm inequalities for positive semi-
definite matrices, Internat. J. Math. 63 (2011), 1121-1138.

  * [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.

  * [32

] J.-C. Bourin and E.-Y. Lee, Decomposition and partial trace of positive
matrices with Hermitian blocks, Internat. J. Math. 24 (2013), no. 1, 1350010,
13 pp.

  * [41

] J.-C. Bourin and E.-Y. Lee, Numerical range and positive block matrices,
Bull. Aust. Math. Soc. 103 (2021), no. 1, 69–77.

  * [45

] J.-C. Bourin, E.-Y. Lee and M. Lin, On a decomposition lemma for positive
semi-definite block-matrices, Linear Algebra Appl. 437 (2012), 1906–1912.

  * [46

] J.-C. Bourin, E.-Y. Lee and M. Lin, Positive matrices partitioned into a
small number of Hermitian blocks, Linear Algebra Appl. 438 (2013), no. 5,
2591–2598.

  * [47

] J.-C. Bourin, A. Mhanna, Positive block matrices and numerical ranges, C. R.
Math. Acad. Sci. Paris 355 (2017), no. 10, 1077–1081.

  * [51

] K. Chen and L.-A. Wu, A matrix realignment method for recognizing
entanglement, Quantum Inf. Comput. 3 (2003), 193-202

  * [53

] Clifford, Applications of Grassmann’ extensive algebra, Amer. Journ. Math. 1
(1878), 350-358.

  * [57

] J. Dereziński, Introduction to representations of the canonical commutation
and anticommutation relations, Lect. Note Phys. 695, 65-145 (2006), Springer.

  * [70

] M. Horodecki, P. Horodecki, R. Horodecki, Separability of mixed states:
necessary and sufficient conditions, Phys. Lett. A 223 (1996) l-8.

  * [84

] M. A. Nielsen and J. Kempe, Separable states are more disordered globally
than locally, Phys. Rev. Lett. 86 (2001) 5184-5187.

  * [86

] D. Petz, Matrix Analysis with some applications,
$<$http://www.math.hu/petz$>$.

  * [88

] S. Ju. Rotfel’d, The singular values of a sum of completely continuous
operators,  Topics in Mathematical Physics, Consultants Bureau, Vol. 3 (1969)
73-78.

###  Chapter 7 Block matrices and numerical range

For sake of completeness the proof of Theorem 7.1.1 is given in Section 7.4
“Around this article”.

Eigenvalue inequalities for positive block matrices with the inradius of the
numerical range [44]

Abstract. We prove the operator norm inequality, for a positive matrix
partitioned into four blocks in $\mathbb{M}_{n}$,

$\left\|\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}\right\|_{\infty}\leq\|A+B\|_{\infty}+\delta(X),$

where $\delta(X)$ is the diameter of the largest possible disc in the
numerical range of $X$. This shows that the inradius
$\varepsilon(X):=\delta(X)/2$ satisfies
$\varepsilon(X)\geq\|X\|_{\infty}-\|(|X^{*}|+|X|)/2\|_{\infty}.$ Several
eigenvalue inequalities are derived. In particular, if $X$ is a normal matrix
whose spectrum lies in a disc of radius $r$, the third eigenvalue of the full
matrix is bounded by the second eigenvalue of the sum of the diagonal block,

$\lambda_{3}\left(\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}\right)\leq\lambda_{2}(A+B)+r.$

We think that $r$ is optimal and we propose a conjecture related to a norm
inequality of Hayashi. Keywords. Numerical range, Partitioned matrices,
eigenvalue inequalities. 2010 mathematics subject classification. 15A60,
47A12, 15A42, 47A30.

####  7.1 Introduction

Positive matrices partitioned into four blocks play a central role in Matrix
Analysis, and in applications, for instance quantum information theory. A lot
of important theorems deal with these matrices. Some of these results give
comparison between the full matrix and its diagonal blocks, in particular the
sum of the diagonal blocks (the partial trace in the quantum terminology).
This note focuses on a recent result of Bourin and Mhana [47], involving the
numerical range of the offdiagonal block. Recall that a symmetric norm
$\|\cdot\|$ on $\mathbb{M}_{2n}$ means a unitarily invariant norm. It induces
a symmetric norm on $\mathbb{M}_{n}$ in an obvious way. The Schatten $p$-norms
$\|\cdot\|_{p}$, $1\leq p\leq\infty$, and the operator norm $(p=\infty)$ are
classical examples of symmetric norms. The main result of [47] reads as
follows.

######  Theorem 7.1.1.

Let $\begin{bmatrix}A&X\\\ X^{*}&B\end{bmatrix}$ be a positive matrix
partitioned into four blocks in $\mathbb{M}_{n}$. Suppose that $W(X)$ has the
width $\omega$. Then, for all symmetric norms,

$\left\|\begin{bmatrix}A&X\\\ X^{*}&B\end{bmatrix}\right\|\leq\|A+B+\omega
I\|.$

Here $I$ stands for the identity matrix, $W(X)$ denotes the numerical range of
$X$, and the width of $W(X)$ is the smallest distance between two parallel
straight lines such that the strip between these two lines contains $W(X)$. If
$\omega=0$, that is $W(X)$ is a line segment, Theorem 1.1 was first proved by
Mhanna [80]. Recently [41], Theorem 7.1.1 has been completed with the reversed
inequality

$\left\|\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}\right\|\geq\left\|\begin{bmatrix}\frac{A+B}{2}+dI&0\\\
0&\frac{A+B}{2}-dI\end{bmatrix}\right\|.$

where $d:=\min\\{|z|\,:\,z\in W(X)\\}$ is the distance from $0$ to $W(X)$.
Several applications were derived.

Some equality cases in Theorem 7.1.1 occur for the operator norm
$\|\cdot\|_{\infty}$ with the following block matrices, where $a,b$ are two
arbitrary nonnegative real numbers.

$\begin{bmatrix}\begin{pmatrix}a&0\\\ 0&b\end{pmatrix}&\begin{pmatrix}0&a\\\
b&0\end{pmatrix}\\\ \begin{pmatrix}0&b\\\
a&0\end{pmatrix}&\begin{pmatrix}b&0\\\ 0&a\end{pmatrix}\end{bmatrix}.$

This follows from the fact that $W\left(\begin{pmatrix}0&b\\\
a&0\end{pmatrix}\right)$ has the width $2\left||a|-|b|\right|$.

Though Theorem 7.1.1 is sharp for the operator norm, a subtle improvement is
possible. This is our concern in the next section. Once again, a geometric
feature of $W(X)$ will contribute: its inradius. Our approach leads to a
remarkable list of eigenvalue that cannot be derived from the norm
inequalities of Theorem 7.1.1. The last section is devoted to some related
operator norm inequalities, in particular we will discuss a property due to
Hayashi (2019) and propose a conjecture.

####  7.2 Eigenvalue inequalities

We define the indiameter $\delta(\Lambda)$ of a compact convex set
$\Lambda\subset\mathbb{C}$ as the diameter of the largest possible disc in
$\Lambda$. For matrices $X\in\mathbb{M}_{n}$, we shorten
$\delta(W(X))=:\delta(X)$. Recall that the numerical range of a two-by-two
matrix is an elliptical disc (or a line segment, or a single point).

A matrix $X\in\mathbb{M}_{n}$ is identified as an operator on
$\mathbb{C}^{n}$. If ${\mathcal{S}}$ is a subspace of $\mathbb{C}^{n}$, we
denote by $X_{\mathcal{S}}$ the compression of $X$ onto ${\mathcal{S}}$. We
then define the elliptical width of $X$ as

$\delta_{2}(X):=\sup_{\dim{\mathcal{S}}=2}\delta(X_{\mathcal{S}}).$

Of course $\delta_{2}(X)\leq\delta(X)\leq\omega$ where $\omega$ still denotes
the width of $W(X)$. If $X$ is a contraction, then $\delta_{2}(X)\leq 1$,
while $\delta(X)$ may be arbitrarily close to 2 (letting $n$ be large enough).
We state our main result.

######  Theorem 7.2.1.

Let $\begin{bmatrix}A&X\\\ X^{*}&B\end{bmatrix}$ be a positive matrix
partitioned into four blocks in $\mathbb{M}_{n}$. Then, for all
$j\in\\{0,1,\ldots,n-1\\}$,

$\lambda_{1+2j}\left(\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}\right)\leq\lambda_{1+j}(A+B)+\delta_{2}(X).$

Here $\lambda_{1}(S)\geq\cdots\geq\lambda_{d}(S)$ stand for the eigenvalue of
any Hermitian matrix $S\in\mathbb{M}_{d}$. If we denote by
$\lambda_{1}^{\uparrow}(S)\leq\cdots\leq\lambda^{\uparrow}_{d}(S)$ these
eigenvalues arranged in the increasing order, then Theorem 7.2.1 reads as

$\lambda_{2k}^{\uparrow}\left(\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}\right)\leq\lambda_{k}^{\uparrow}(A+B)+\delta_{2}(X).$

for all $k\in\\{1,2,\ldots,n\\}$.

The case $j=0$ in Theorem 7.2.1 improves Theorem 7.1.1 for the operator norm.
We may consider that Theorem 7.2.1 is trivial for $j=n-1$. Indeed, using the
decomposition [31, Lemma 3.4],

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

for some unitary matrices $U,\,V\in\mathbb{M}_{2n}$, we obtain from Weyl’s
inequality [13, p. 62],

$\displaystyle\lambda_{2n-1}\left(\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}\right)$
$\displaystyle\leq\lambda_{n}\left(\begin{bmatrix}A&0\\\
0&0\end{bmatrix}\right)+\lambda_{n}\left(\begin{bmatrix}0&0\\\
0&B\end{bmatrix}\right)$ $\displaystyle=\lambda_{n}(A)+\lambda_{n}(B)$
$\displaystyle\leq\lambda_{n}(A+B).$

We turn to the proof of the theorem.

###### Proof.

We first consider the case $j=0$. We may assume that the norm of the block
matrix is strictly greater than the norms of its two diagonal blocks $A$ and
$B$, otherwise the statement is trivial. Hence we have two nonzero (column)
vectors $h_{1},h_{2}\in\mathbb{C}^{n}$ such that
$\|h_{1}\|^{2}+\|h_{2}\|^{2}=1$ and

$\lambda_{1}\left(\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}\right)=\left\|\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}\right\|_{\infty}=\begin{pmatrix}h^{*}_{1}&h_{2}^{*}\end{pmatrix}\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}\begin{pmatrix}h_{1}\\\ h_{2}\end{pmatrix}.$

Therefore, denoting by $E_{1}$ and $E_{2}$ the rank one projections
corresponding to the one dimensional subspaces spanned by $h_{1}$ and by
$h_{2}$, we have

$\left\|\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}\right\|_{\infty}=\left\|\begin{bmatrix}E_{1}&0\\\
0&E_{2}\end{bmatrix}\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}\begin{bmatrix}E_{1}&0\\\
0&E_{2}\end{bmatrix}\right\|_{\infty}$

Hence, denoting by $F$ a rank two projection such that $E_{1}\leq F$ and
$E_{2}\leq F$, we have

$\displaystyle\left\|\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}\right\|_{\infty}$
$\displaystyle=\left\|\begin{bmatrix}F&0\\\
0&F\end{bmatrix}\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}\begin{bmatrix}F&0\\\ 0&F\end{bmatrix}\right\|_{\infty}$
$\displaystyle=\left\|\begin{bmatrix}FAF&FXF\\\
FX^{*}F&FBF\end{bmatrix}\right\|_{\infty}.$

So, letting ${\mathcal{S}}$ denote the range of $F$, we have

$\left\|\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}\right\|_{\infty}=\left\|\begin{bmatrix}A_{\mathcal{S}}&X_{\mathcal{S}}\\\
X^{*}_{\mathcal{S}}&B_{\mathcal{S}}\end{bmatrix}\right\|_{\infty}.$

Hence applying Theorem 7.1.1 for the operator norm, we obtain

$\left\|\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}\right\|_{\infty}\leq\|A_{\mathcal{S}}+B_{\mathcal{S}}\|_{\infty}+\varepsilon$

where $\varepsilon$ is the width of $W(X_{\mathcal{S}})$. Since
$W(X_{\mathcal{S}})$ is an elliptical disc (as $X_{\mathcal{S}}$ acts on a
two-dimensional space), its width equals to its indiameter, hence
$\varepsilon\leq\delta_{2}(X)$, and since

$\|A_{\mathcal{S}}+B_{\mathcal{S}}\|_{\infty}=\|(A+B)_{\mathcal{S}}\|_{\infty}\leq\|A+B\|_{\infty}=\lambda_{1}(A+B),$

the proof for $j=0$ is complete.

We turn to the general case, $j=1,\ldots,n-1$. By the min-max principle,

$\displaystyle\lambda_{1+2j}\left(\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}\right)$
$\displaystyle\leq\inf_{\dim{\mathcal{S}}=n-j}\lambda_{1}\left(\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}_{{\mathcal{S}}\oplus{\mathcal{S}}}\right)$
$\displaystyle=\inf_{\dim{\mathcal{S}}=n-j}\lambda_{1}\left(\begin{bmatrix}A_{\mathcal{S}}&X_{\mathcal{S}}\\\
X^{*}_{\mathcal{S}}&B_{\mathcal{S}}\end{bmatrix}\right),$

hence, from the first part of the proof,

$\displaystyle\lambda_{1+2j}\left(\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}\right)$
$\displaystyle\leq\inf_{\dim{\mathcal{S}}=n-j}\lambda_{1}\left(A_{\mathcal{S}}+B_{\mathcal{S}}\right)+\delta_{2}(X)$
$\displaystyle=\lambda_{1+j}(A+B)+\delta_{2}(X)$

which is the desired claim. ∎

If $X\in\mathbb{M}_{n}$, we denote by ${\mathrm{dist}}(X,\mathbb{C}I)$ the
$\|\cdot\|_{\infty}$-distance from $X$ to $\mathbb{C}I$. Thus, for a scalar
perturbation of a contraction, $X=\lambda I+C$ for some contraction
$C\in\mathbb{M}_{n}$ and some $\lambda\in\mathbb{C}$, we have
${\mathrm{dist}}(X,\mathbb{C}I)\leq 1$.

######  Corollary 7.2.2.

Let $\begin{bmatrix}A&X\\\ X^{*}&B\end{bmatrix}$ be a positive matrix
partitioned into four blocks in $\mathbb{M}_{n}$. Then, for all
$j\in\\{0,1,\ldots,n-1\\}$,

$\lambda_{1+2j}\left(\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}\right)\leq\lambda_{1+j}(A+B)+{\mathrm{dist}}(X,\mathbb{C}I).$

###### Proof.

For any subspace ${\mathcal{S}}\subset C^{n}$, we have

${\mathrm{dist}}(X,\mathbb{C}I)\geq{\mathrm{dist}}(X_{\mathcal{S}},\mathbb{C}I_{\mathcal{S}}).$

If ${\mathcal{S}}$ has dimension 2, then

${\mathrm{dist}}(X_{\mathcal{S}},\mathbb{C}I_{\mathcal{S}})\geq\delta\left(W(X_{\mathcal{S}})\right).$

Therefore ${\mathrm{dist}}(X,\mathbb{C}I)\geq\delta_{2}(X)$ and Theorem 7.2.1
completes the proof. ∎

######  Corollary 7.2.3.

Let $A,B\in\mathbb{M}_{n}$. Then, for every $j\geq 0$ such that $1+2j\leq n$,

$\lambda_{1+2j}\left(A^{*}A+B^{*}B\right)\leq\lambda_{1+j}\left(AA^{*}+BB^{*}\right)+\delta_{2}(AB^{*})$

###### Proof.

Note that

$\lambda_{1+2j}\left(A^{*}A+B^{*}B\right)=\lambda_{1+2j}\left(T^{*}T\right)=\lambda_{1+2j}\left(TT^{*}\right)$

with $T=\begin{bmatrix}A\\\ B\end{bmatrix}$ and
$TT^{*}=\begin{bmatrix}AA^{*}&AB^{*}\\\ BA^{*}&BB^{*}\end{bmatrix}$ so that
Theorem 7.2.1 yields the desired claim. ∎

######  Corollary 7.2.4.

Let $\begin{bmatrix}A&N\\\ N^{*}&B\end{bmatrix}$ be a positive matrix
partitioned into four blocks in $\mathbb{M}_{n}$. If $N$ is normal and its
spectrum is contained in a disc of radius $r$, then,

$\lambda_{1+2j}\left(\begin{bmatrix}A&N\\\
N^{*}&B\end{bmatrix}\right)\leq\lambda_{1+j}(A+B)+r.$

for all $j=0,1,\ldots,n-1$.

###### Proof.

Corollary 7.2.4 is a special case of corollary 7.2.2, as $N=\lambda I+R$,
where $\lambda$ is the center of the disc of radius $r$ containing the
spectrum of $N$, and $\|N\|_{\infty}\leq r$. ∎

######  Question 7.2.5.

Fix $r>0$ and $\varepsilon>0$. Can we find (with $n$ large enough) a normal
matrix $N$ with spectrum in a disc of radius $r$ and a positive block matrix
$\begin{bmatrix}A&N\\\ N^{*}&B\end{bmatrix}$ such that

$\lambda_{1+2j}\left(\begin{bmatrix}A&N\\\
N^{*}&B\end{bmatrix}\right)\geq\lambda_{1+j}(A+B)+r-\varepsilon$

for some $j\in\\{0,\ldots,n-1\\}$ ? Is it true for for $j=0$ ?

####  7.3 Norm inequalities

Corollary 7.2.4 with $j=0$ reads as follows.

######  Corollary 7.3.1.

Let $\begin{bmatrix}A&N\\\ N^{*}&B\end{bmatrix}$ be a positive matrix
partitioned into four blocks in $\mathbb{M}_{n}$. If $N$ is normal and its
spectrum is contained in a disc of radius $r$, then,

$\left\|\begin{bmatrix}A&N\\\
N^{*}&B\end{bmatrix}\right\|_{\infty}\leq\|A+B\|_{\infty}+r.$

We do not know wether the constant $r$ is sharp or not (Question 7.2.5). If
$n=2$, we can replace $r$ by $0$ as the numerical range of $N$ is then a line
segment. If $n=3$ there are some simple examples with $N=U$ unitary such that

$\left\|\begin{bmatrix}A&U\\\
U^{*}&B\end{bmatrix}\right\|_{\infty}>\|A+B\|_{\infty}.$

See Hayashi’s example in the discussion of [H, Problem 3] and the interesting
study and examples in [G] where we further have $A+B=kI$ for some scalars $k$.
The next result is due to Hayashi [H, Theorem 2.5].

######  Theorem 7.3.2.

Suppose that $X\in\mathbb{M}_{n}$ is invertible with $n$ distinct singular
values. If the inequality

$\left\|\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}\right\|_{\infty}\leq\|A+B\|_{\infty}.$

holds for all positive block-matrix with $X$ as off-diagonal block, then $X$
is normal.

Theorem 7.3.2 and Theorem 7.1.1 suggest a natural conjecture. If $W(T)$ is
line segment, then $T$ is a so-called essentially Hermitian matrix.

######  Conjecture 7.3.3.

Let $X\in\mathbb{M}_{n}$. If the inequality

$\left\|\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}\right\|_{\infty}\leq\|A+B\|_{\infty}$

holds for all positive block-matrix with $X$ as off-diagonal block, then $X$
is essentially Hermitian.

If we replace the operator norm by the Frobenius (or Hilbert-Schmidt) norm
$\|\cdot\|_{2}$ then the following characterization holds.

######  Proposition 7.3.4.

Let $X\in\mathbb{M}_{n}$. Then, the inequality

$\left\|\begin{bmatrix}A&X\\\ X^{*}&B\end{bmatrix}\right\|_{2}\leq\|A+B\|_{2}$

holds for all positive block-matrix with $X$ as off-diagonal block if and only
if $X$ is normal.

###### Proof.

Suppose that $X$ is normal. To prove the inequality, squaring both side, it
suffices to establish the trace inequality

${\mathrm{Tr\,}}X^{*}X\leq{\mathrm{Tr\,}}AB.$ (7.3.1)

Note that $X=A^{1/2}KB^{1/2}$, for some contraction $K$. Recall that, for all
symmetric norms on $\mathbb{M}_{n},$ and any normal matrix $N\mathbb{M}_{n}$,
decomposed as $N=ST$, we have $\|N\|\leq\|TS\|$. Therefore

$\|X\|=\|A^{1/2}KB^{1/2}\|\leq\|KB^{1/2}A^{1/2}\|.$

Squaring this inequality with the Frobenius norm yields the desired inequality
(7.3.1).

Suppose that $X$ is nonnormal, and note that $\begin{bmatrix}|X^{*}|&X\\\
X^{*}&|X|\end{bmatrix}$ is positive semidefinite and satisfies

$\left\|\begin{bmatrix}|X^{*}|&X\\\
X^{*}&|X|\end{bmatrix}\right\|_{2}^{2}=4\||X|\|_{2}^{2}$

while

$\||X^{*}|+|X|\|_{2}^{2}=2\||X|\|_{2}^{2}+2{\mathrm{Tr\,}}|X||X^{*}|$

In the Hilbert space $(\mathbb{M}_{n},\|\cdot\|_{2})$, the assumption

$\||X|\|_{2}=\||X^{*}|\|_{2},\quad|X|\neq|X^{*}|$

ensures strict inequality in the Cauchy-Schwarz inequality

${\mathrm{Tr\,}}|X||X^{*}|<\|X\|_{2}^{2}.$

Therefore

$\||X^{*}|+|X|\|_{2}^{2}<\left\|\begin{bmatrix}|X^{*}|&X\\\
X^{*}&|X|\end{bmatrix}\right\|_{2}^{2}$

and this completes the proof. ∎

Proposition 7.3.4 suggests a question: for which $p\in[1,\infty]$, the
schatten $p$-norm inequality

$\left\|\begin{bmatrix}A&N\\\ N^{*}&B\end{bmatrix}\right\|_{p}\leq\|A+B\|_{p}$

holds for any positive partitioned matrices with a normal off-diagonal block
$N$ ?

######  Corollary 7.3.5.

Let $H,K,X\in\mathbb{M}_{n}$ be Hermitian. If $X$ is invertible and $HK$ is a
scalar perturbation of a contraction, then,

$\left\|XH^{2}X+X^{-1}K^{2}X^{-1}\right\|_{\infty}\leq\left\|HX^{2}H+KX^{-2}K\right\|_{\infty}+1.$

###### Proof.

We apply Corollary 7.2.3 with $j=0$ and $A=HX$, $B=KX^{-1}$, to get

$\left\|XH^{2}X+X^{-1}K^{2}X^{-1}\right\|_{\infty}\leq\left\|HX^{2}H+KX^{-2}K\right\|_{\infty}+\delta_{2}(HK)$

Since $(HK)_{\mathcal{S}}$ is a scalar perturbation of a contraction acting on
a space of dimension 2, necessarily $\delta_{2}(HK)\leq 1$. ∎

For a normal operator, the numerical range is the convex hull of the spectrum.
For a non normal operator $X$, several lower bounds for the indiameter of
$W(X)$ can be obtained from the left and right modulus $|X^{*}|$ and $|X|$.

######  Corollary 7.3.6.

Let $X\in\mathbb{M}_{n}$ and let $f(t)$ and $g(t)$ are two nonnegative
functions defined on $[0,\infty)$ such that $f(t)g(t)=t^{2}$. Then,

$\delta_{2}(X)\geq\left\|f(|X|)+g(|X|)\right\|_{\infty}-\left\|f(|X^{*}|)+g(|X|)\right\|_{\infty}.$

###### Proof.

First, observe that we have a function $h(t)$ defined on $[0,\infty)$ such
that

$f(t)=th^{2}(t^{1/2}),\quad g(t)=th^{-2}(t^{1/2}),$ (7.3.2)

and $h(t)>0$ for all $t\geq 0$ (we may, for instance, set $h(0)=1$). Hence
$h(T)$ is invertible for any positive $T$, and from the polar decomposition

$X=|X^{*}|^{1/2}U|X|^{1/2}$

with a unitary factor $U$, we infer the factorization

$X=|X^{*}|^{1/2}h(|X^{*}|^{1/2})U|X|^{1/2}h^{-1}(|X|^{1/2}).$

Thus $X=AB^{*}$ where $A=|X^{*}|^{1/2}h(|X^{*}|^{1/2})$ and
$B^{*}=U|X|^{1/2}h^{-1}(|X|^{1/2})$. Therefore Corollary 7.2.3 yields

$\left\||X^{*}|h^{2}(|X^{*}|^{1/2})+U|X|h^{-2}(|X|^{1/2})U^{*}\right\|_{\infty}\leq\left\||X|h^{2}(|X|^{1/2})+|X|h^{-2}(|X|^{1/2})U^{*}\right\|_{\infty}+\delta_{2}(X)$

Using (7.3.2) and the fact that $\varphi(|X^{*}|)=U\varphi(|X|)U^{*}$ for any
function $\varphi(t)$ defined on $[0,\infty)$, the proof is complete. ∎

The following special case shows that Corollary 7.3.6 is rather optimal.

######  Corollary 7.3.7.

If $X\in\mathbb{M}_{n}$ has a numerical range of inradius $\varepsilon(X)$,
then, for all $a\in\mathbb{C}$,

$\varepsilon(X)\geq\left\|X-aI\right\|_{\infty}-\left\|\frac{|X-aI|+|X^{*}-\overline{a}I|}{2}\right\|_{\infty}.$

If $X\in\mathbb{M}_{2}$ and $a=\tau$ is the normalized trace of $X$, then this
inequality is an equality.

###### Proof.

Applying Corollary 7.3.6 with $X-aI$ and $f(t)=g(t)=t$ yields the inequality.
If $X\in\mathbb{M}_{2}$, then $X$ is unitarily equivalent to

$\begin{pmatrix}\tau&y\\\ x&\tau\end{pmatrix}.$

So

$\displaystyle\|X-\tau I\|_{\infty}-\left\|\frac{|X-\tau
I|+|X^{*}-\overline{\tau}I|}{2}\right\|_{\infty}$
$\displaystyle=\left\|\begin{pmatrix}|x|&0\\\
0&|y|\end{pmatrix}\right\|_{\infty}-\frac{1}{2}\left\|\begin{pmatrix}|x|+|y|&0\\\
0&|x|+|y|\end{pmatrix}\right\|_{\infty}$ $\displaystyle=\left||x|-|y|\right|$
$\displaystyle=\varepsilon(X)$

establishing the desired equality. ∎

The special case $f(t)=g(t)=t$ in Corollary 7.3.6 seems important, we record
it as a proposition:

######  Proposition 7.3.8.

The elliptical width of the numerical range of $X\in\mathbb{M}_{n}$ satisfies

$\delta_{2}(X)\geq 2\|X\|_{\infty}-\||X|+|X^{*}|\|_{\infty}.$

In particular, the inradius $\varepsilon(X)$ of the numerical range of $X$
satisfies

$\varepsilon(X)\geq\|X\|_{\infty}-\|(|X|+|X^{*}|)/2\|_{\infty}.$

######  Remark 7.3.9.

Our results still hold for operators on infinite dimensional separable Hilbert
space (assuming in Corollary 7.3.6 that $f(t)$ and $g(t)$ are Borel
functions).

####  7.4 Around this article

We give the proof of Theorem 7.1.1.

###### Proof.

(Theorem 7.1.1) By using the unitary congruence implemented by

$\begin{bmatrix}e^{i\theta}&0\\\ 0&I\end{bmatrix}$

we see that our block matrix is unitarily equivalent to

$\begin{bmatrix}A&e^{i\theta}X\\\ e^{-i\theta}X^{*}&B\end{bmatrix}.$

As $W(e^{i\theta}X)=e^{i\theta}W(X)$, by choosing the adequate $\theta$ and
replacing $X$ by $e^{i\theta}X$, we may and do assume that $W(X)$ lies in a
strip ${\mathcal{S}}$ of width $\omega$ and parallel to the imaginary axis,

${\mathcal{S}}=\left\\{\,x+iy\ :\quad y\in\mathbb{R},\quad r\leq x\leq
r+\omega\,\right\\}.$

The projection property for the real part
${\mathrm{Re}\,}W(X)=W({\mathrm{Re}\,}X)$, then ensures that

$rI\leq{\mathrm{Re}\,}X\leq(r+\omega)I.$ (7.4.1)

Now we use the decomposition [45, Corollary 2.1] derived from (8.2.1),

$\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}=U\begin{bmatrix}\frac{A+B}{2}+{\mathrm{Re}\,}X&0\\\
0&0\end{bmatrix}U^{*}+V\begin{bmatrix}0&0\\\
0&\frac{A+B}{2}-{\mathrm{Re}\,}X\end{bmatrix}V^{*}$ (7.4.2)

for some unitaries $U,\,V\in\mathbb{M}_{2n}$. Note that the two matrices in
the right hand side of (7.4.2) are positive since so are

$\begin{bmatrix}I&I\end{bmatrix}\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}\begin{bmatrix}I\\\
I\end{bmatrix}\quad{\mathrm{and}}\begin{bmatrix}I&-I\end{bmatrix}\begin{bmatrix}A&X\\\
X^{*}&B\end{bmatrix}\begin{bmatrix}I\\\ -I\end{bmatrix}.$

Combining (7.4.1) and (7.4.2) yields

$\begin{bmatrix}A&X\\\ X^{*}&B\end{bmatrix}\leq
U\begin{bmatrix}\frac{A+B}{2}+(r+\omega)I&0\\\
0&0\end{bmatrix}U^{*}+V\begin{bmatrix}0&0\\\
0&\frac{A+B}{2}-rI\end{bmatrix}V^{*}$

where the two matrices of the right hand side are positive. From each Ky Fan
$k$-norm, $k=1,2,\ldots,2n$, we then have

$\displaystyle\left\|\begin{bmatrix}A&X\\\ X^{*}&B\end{bmatrix}\right\|_{(k)}$
$\displaystyle\leq\left\|\frac{A+B}{2}+(r+\omega)I\right\|_{(k)}+\left\|\frac{A+B}{2}-rI\right\|_{(k)}$
$\displaystyle=\left\|A+B+\omega I\right\|_{(k)}.$

The Ky Fan principle then guarantees that this inequality hold for all
symmetric norms. ∎

####  7.5 References of Chapter 7

  * [G

] M. Gumus, J. Liu, S. Raouafi, T-Y. Tam, Positive semi-definite $2\times 2$
block matrices and norm inequalities, Linear Algebra Appl. 551 (2018), 83–91.

  * [H

] T. Hayashi, On a norm inequality for a positive block-matrix, Linear Algebra
Appl. 566 (2019), 86–97.

  * [41

] J.-C. Bourin and E.-Y. Lee, Numerical range and positive block matrices,
Bull. Aust. Math. Soc. 103 (2021), no. 1, 69–77.

  * [44

] J.-C. Bourin and E.-Y. Lee, Eigenvalue inequalities for positive block
matrices with the inradius of the numerical range, preprint

  * [45

] J.-C. Bourin, E.-Y. Lee and M. Lin, On a decomposition lemma for positive
semi-definite block-matrices, Linear Algebra Appl. 437 (2012), 1906–1912.

  * [47

] J.-C. Bourin, A. Mhanna, Positive block matrices and numerical ranges, C. R.
Math. Acad. Sci. Paris 355 (2017), no. 10, 1077–1081.

  * [80

] A. Mhanna, On symmetric norm inequalities and positive definite block-
matrices, Math. Inequal. Appl. 21 (2018), no. 1, 133–138.

###  Chapter 8 Block matrices and Pythagoras

A Pythagorean theorem for partitioned matrices [42]

Abstract. We establish a Pythagorean theorem for the absolute values of the
blocks of a partitioned matrix. This leads to a series of remarkable operator
inequalities. For instance, if the matrix $\mathbb{A}$ is partitioned into
three blocks $A,B,C$, then

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

for some isometries $U,V,W$, and

$\mu_{4}^{2}(\mathbb{A})\leq\mu_{3}^{2}(A)+\mu_{2}^{2}(B)+\mu_{1}^{2}(C)$

where $\mu_{j}$ stands for the $j$-th singular value. Our theorem may be used
to extend a result by Bhatia and Kittaneh for the Schatten $p$-noms and to
give a singular value version of Cauchy’s Interlacing Theorem.

Keywords. Partitioned matrices, functional calculus, matrix inequalities. 2010
mathematics subject classification. 15A18, 15A60, 47A30.

####  8.1 Introduction

Let $\mathbb{M}_{d}$ denote the space of $d$-by-$d$ matrices. If
$\mathbb{A}\in\mathbb{M}_{d}$, the polar decomposition holds,

$\mathbb{A}=U|\mathbb{A}|$ (8.1.1)

where $|\mathbb{A}|\in\mathbb{M}_{d}$ is positive semi-definite and
$U\in\mathbb{M}_{d}$ is a unitary matrix. The matrix $|\mathbb{A}|$ is called
the absolute value of $\mathbb{A}$, and its eigenvalues are the singular
values of $\mathbb{A}$. The absolute value can be defined for $d\times
d^{\prime}$ matrices $\mathbb{A}\in\mathbb{M}_{d,d^{\prime}}$ as a positive
matrix $|\mathbb{A}|\in\mathbb{M}_{d^{\prime}}$, and the factor $U$ in (8.1.1)
is an isometry ($d\geq d^{\prime}$) or a coisometry ($d<d^{\prime}$).

If $\mathbb{A}$ is partitioned in some number of rectangular blocks, say four
blocks $A,B,C,D$, it is of interest to have a relation between the absolute
value $|\mathbb{A}|$ and the absolute values of the blocks. By using the
standard inner product of $\mathbb{M}_{d,d^{\prime}}$, we immediately have the
trace relation

${\mathrm{Tr\,}}|\mathbb{A}|^{2}={\mathrm{Tr\,}}|A|^{2}+{\mathrm{Tr\,}}|B|^{2}+{\mathrm{Tr\,}}|C|^{2}+{\mathrm{Tr\,}}|D|^{2}.$

This note aims to point out a much stronger Pythagorean theorem, Theorem
8.2.1, and several consequences. This result holds for many partitionings of
$\mathbb{A}$, not only when $A$ is a block matrix partitioned into $p\times q$
blocks. For instance, one may consider the matrix