Norm inequalities involving a special class of functions for sector matrices

Abstract

In this paper, we present some unitarily invariant norm inequalities for sector matrices involving a special class of functions. In particular, if $Z=\left(\begin{array}{cc}{Z}_{11}& Z_{12}\\ Z_{21}& Z_{22}\end{array}\right)$ is a $2n\times 2n$ matrix such that numerical range of Z is contained in a sector region $S_{\alpha }$ for some $\alpha \in [0,\frac{\pi }{2} )$, then, for a submultiplicative function h of the class $\mathcal{C}$ and every unitarily invariant norm, we have

$$\begin{aligned} \bigl\Vert h \bigl( \vert Z_{ij} \vert ^{2} \bigr) \bigr\Vert &\leq \bigl\Vert h^{r} \bigl( \sec (\alpha ) \vert Z_{11} \vert \bigr) \bigr\Vert ^{\frac{1}{r} } \bigl\Vert h^{s} \bigl( \sec (\alpha ) \vert Z_{22} \vert \bigr) \bigr\Vert ^{ \frac{1}{s} }, \end{aligned}$$

where r and s are positive real numbers with $\frac{1}{r}+\frac{1}{s}=1$ and $i,j=1,2$. We also extend some unitarily invariant norm inequalities for sector matrices.

1 Introduction and preliminaries

Let ${\mathcal{M}}_{n}$ be the algebra of all $n\times n$ complex matrices. For $Z\in {\mathcal{M}}_{n}$, the conjugate transpose of Z is denoted by $Z^{*}$. A complex matrix $Z\in {\mathcal{M}}_{2n}$ can be partitioned as a $2 \times 2$ block matrix

$$ Z= \begin{pmatrix} Z_{11} &Z_{12} \cr Z_{21} & Z_{22} \end{pmatrix}, $$

(1)

where $Z_{ij}\in {\mathcal{M}}_{n}$ ($i,j=1,2$). For $Z\in {\mathcal{M}}_{n}$, let $Z ={\mathcal{R}e}(Z)+ i{\mathcal{I}m}(Z)$ be the Cartesian decomposition of Z, where the Hermitian matrices ${\mathcal{R}e}(Z)=\frac{Z+Z^{*}}{2}$ and ${\mathcal{I}m}(Z)=\frac{Z-Z^{*}}{2i}$ are called the real and imaginary parts of Z, respectively. We say that a matrix $Z\in {\mathcal{M}}_{n}$ is positive semidefinite if $z^{*}Zz\geq 0$ for all complex numbers z. For $Z \in {\mathcal{M}}_{n}$, let $s_{1}(Z) \geq s_{2}(Z) \geq \cdots \geq s_{n}(Z)$ denote the singular values of Z, i.e. the eigenvalues of the positive semidefinite matrix $\vert Z \vert = (Z^{*}Z)^{\frac{1}{2}}$ arranged in a decreasing order and repeated according to multiplicity. Note that $s_{j}(Z) = s_{j}(Z^{*}) = s_{j}(\vert Z \vert )$ for $j = 1, 2,\ldots , n$. A norm $\Vert \cdot \Vert$ on ${\mathcal{M}}_{n}$ is said to be unitarily invariant if $\Vert UZV \Vert = \Vert Z \Vert$ for every $Z \in {\mathcal{M}}_{n}$ and for every unitary $U, V \in {\mathcal{M}}_{n}$. For $Z\in {\mathcal{M}}_{n}$ and $p>0$, let $\Vert Z \Vert _{p}= ( \sum_{j=1}^{n}s_{j}^{p}(Z) )^{\frac{1}{p}}$. This defines the Schatten p-norm (quasinorm) for $p\geq 1$ ($0< p<1$). It is clear that the Schatten p-norm is an unitarily invariant norm. The w-norm of a matrix $Z\in {\mathcal{M}}_{n}$ is defined by $\Vert Z\Vert _{w}=\sum_{j=1}^{n} w_{j} s_{j} (Z)$, where $w=(w_{1},w_{2},\ldots ,w_{n})$ is a decreasing sequence of nonnegative real numbers.

In this paper, we assume that all functions are continuous. It is known that if $Z\in {\mathcal{M}}_{n}$ is positive semidefinite and h is a nonnegative increasing function on $[0, \infty )$, then $h(s_{j}(Z)) = s_{j} (h(Z) )$ for $j = 1, 2, \ldots , n$. For positive semidefinite $X, Y \in {\mathcal{M}}_{n}$ and a nonnegative increasing function h on $[0, \infty )$, if $s_{j}(X)\leq s_{j}(Y)$ for $j = 1, 2, \ldots , n$, then $\Vert h(X) \Vert \leq \Vert h(Y) \Vert$, where $\Vert \cdot \Vert$ is a unitarily invariant norm. For more information, see [4, 18] and references therein.

We say that a matrix Z is accretive (respectively dissipative) if in the Cartesian decomposition $Z=X+iY$, the matrix X (respectively Y) is positive semidefinite. If both X and Y are positive semidefinite, Z is called accretive–dissipative.

Another important class of matrices, which is related to the class of accretive–dissipative matrices, is called sector matrices. To introduce this class, let $\alpha \in [0,\frac{\pi }{2} )$ and $S_{\alpha }$ be a sector defined in the complex plane by

$$ S_{\alpha } = \bigl\lbrace z \in C : {\mathcal{R}e}(z) \geq 0, \bigl\vert {\mathcal{I}m}(z) \bigr\vert \leq \tan (\alpha ) {\mathcal{R}e}(z) \bigr\rbrace . $$

For $Z\in {\mathcal{M}}_{n}$, the numerical range of Z is defined by

$$ W(A)=\bigl\lbrace z^{*}Zz : z\in C , \Vert z \Vert =1\bigr\rbrace . $$

A matrix whose its numerical range is contained in a sector region $S_{\alpha }$ for some $\alpha \in [0,\frac{\pi }{2} )$, is called a sector matrix. It follows from the definition of sector matrices that Z is positive semidefinite if and only if $W(Z) \subseteq S_{0}$ and also Z is accretive–dissipative if and only if $W(e^{\frac{-i\pi }{4}}Z) \subseteq S_{\frac{\pi }{4}}$. Moreover, if $W(Z) \subseteq S_{\alpha }$, then Z is invertible with ${\mathcal{R}e}(Z)>0$ and therefore Z is accretive. For more on sector matrices see [3, 6, 7, 11–15, 17, 19–22] and the references therein. For $x=(x_{1},x_{2},\ldots ,x_{n})$ and $y=(y_{1},y_{2},\ldots ,y_{n})\in R^{n}$ with nonnegative components, if $\sum_{j=1}^{k} x_{j} \leq \sum_{j=1}^{k} y_{j}$ ($\prod_{j=1}^{k} x_{j} \leq \prod_{j=1}^{k} y_{j}$) for $k=1, 2,\ldots , n$, then we say that x is weakly (weakly log) majorized by y and denoted by $x\prec _{\omega } y ( x\prec _{\omega \log } y )$. It is known that weak log majorization implies weak majorization. A nonnegative function h on the interval $[0, \infty )$ is said to be submultiplicative if $h(ab) \leq h(a)h(b)$ whenever $a, b\in [0, \infty )$.

Gumus et al. [8] introduced the special class $\mathcal{C}$ involving all nonnegative increasing functions h on $[0, \infty )$ satisfying the following condition: If $x =(x_{1},x_{2},\ldots ,x_{n})$ and $y=(y_{1},y_{2},\ldots ,y_{n})$ are two decreasing sequences of nonnegative real numbers such that $\prod_{j=1}^{k} x_{j} \leq \prod_{j=1}^{k} y_{j}$ ($k=1, 2, \ldots , n$), then $\prod_{j=1}^{k} h(x_{j}) \leq \prod_{j=1}^{k} h(y_{j})$ ($k=1, 2, \ldots , n$).

Note that the power function $h(t)=t^{p}$ ($p>0$) belongs to class $\mathcal{C}$. For more information about the class $\mathcal{C}$ see [8] and the references therein. For the positive semidefinite matrix $\left(\begin{array}{cc}X& Z\\ Z^{\ast }& Y\end{array}\right)\in {\mathcal{M}}_{2n}$, one proved [8] that, if $h \in \mathcal{C}$ is a submultiplicative function, then

$$ \bigl\Vert h \bigl( \vert Z \vert ^{2} \bigr) \bigr\Vert \leq \bigl\Vert h^{r} ( X ) \bigr\Vert ^{\frac{1}{r} } \bigl\Vert h^{s} (Y ) \bigr\Vert ^{\frac{1}{s} }, $$

(2)

where r and s are positive real numbers with $\frac{1}{r}+\frac{1}{s}=1$. Furthermore, for accretive–dissipative matrix $Z\in {\mathcal{M}}_{2n}$ partitioned as in (1), one showed the following unitarily invariant norm inequalities:

$$ \bigl\Vert h \bigl( \vert Z_{12} \vert ^{2} \bigr) + h \bigl( \bigl\vert Z_{21}^{*} \bigr\vert ^{2} \bigr) \bigr\Vert \leq \bigl\Vert h^{r} \bigl( 2 \vert Z_{11} \vert \bigr) \bigr\Vert ^{\frac{1}{r} } \bigl\Vert h^{s} \bigl( 2 \vert Z_{22} \vert \bigr) \bigr\Vert ^{\frac{1}{s} }, $$

(3)

where $h \in \mathcal{C}$ is a submultiplicative convex function and

$$ \bigl\Vert h \bigl( \vert Z_{12} \vert ^{2} \bigr) + h \bigl( \bigl\vert Z_{21}^{*} \bigr\vert ^{2} \bigr) \bigr\Vert \leq 4 \bigl\Vert h^{r} \bigl( \vert Z_{11} \vert \bigr) \bigr\Vert ^{\frac{1}{r} } \bigl\Vert h^{s} \bigl( \vert Z_{22} \vert \bigr) \bigr\Vert ^{\frac{1}{s} }, $$

(4)

where $h \in \mathcal{C}$ is a submultiplicative concave function such that r and s are positive real numbers with $\frac{1}{r}+\frac{1}{s}=1$. Moreover, for a sector matrix $Z\in {\mathcal{M}}_{2n}$ partitioned as in (1), Zhang [22] proved the following inequality:

$$ \Vert Z_{12} \Vert ^{2} \leq \sec ^{2}(\alpha ) \Vert Z_{11} \Vert \Vert Z_{22} \Vert $$

(5)

for any unitarily invariant norm and $\alpha \in [0,\frac{\pi }{2} )$. Alakhrass [1] extended inequality (5) to

$$ \Vert \vert Z_{12} \vert ^{p} \Vert \leq \sec ^{p}(\alpha ) \bigl\Vert Z_{11}^{ \frac{pr}{2}} \bigr\Vert ^{\frac{1}{r} } \bigl\Vert Z_{22}^{\frac{ps}{2}} \bigr\Vert ^{ \frac{1}{s} }, $$

(6)

where r, s and p are positive numbers in which $\frac{1}{r}+\frac{1}{s}=1$ and $\alpha \in [0,\frac{\pi }{2} )$.

In [8], the authors presented some Schatten p-norm inequalities for accretive–dissipative matrices $Z\in {\mathcal{M}}_{2n}$ partitioned as in (1), which compared the off-diagonal blocks of Z to its diagonal blocks as follows:

$$ \Vert Z_{12} \Vert _{p}^{p}+ \Vert Z_{21} \Vert _{p}^{p}\leq 2^{p-1} \Vert Z_{11} \Vert _{p}^{\frac{p}{2}} \Vert Z_{22} \Vert _{p}^{ \frac{p}{2}}\quad (p\geq 2) $$

(7)

and

$$ \Vert Z_{12} \Vert _{p}^{p}+ \Vert Z_{21} \Vert _{p}^{p}\leq 2^{3-p} \Vert Z_{11} \Vert _{p}^{\frac{p}{2}} \Vert Z_{22} \Vert _{p}^{ \frac{p}{2}}\quad (0< p\leq 2). $$

(8)

Let $Z_{ij}$ ($1\leq i,j \leq n$) be square matrices of the same size such that the block matrix

$$ Z= \begin{pmatrix} Z_{11} &Z_{12} &\cdots & Z_{1n} \cr Z_{21} &Z_{22} &\cdots & Z_{2n} \cr \vdots &\vdots &\cdots &\vdots \cr Z_{n1} & Z_{n2} &\cdots & Z_{nn} \end{pmatrix} $$

(9)

be accretive–dissipative. For such matrices, Kittaneh and Sakkijha [10] showed that

$$ \sum_{i\neq j} \Vert Z_{ij} \Vert _{p}^{p}\leq (n-1)2^{p-2}\sum_{i=1}^{n} \Vert Z_{ii} \Vert _{p}^{p}\quad (p\geq 2) $$

(10)

and

$$ \sum_{i\neq j} \Vert Z_{ij} \Vert _{p}^{p}\leq (n-1)2^{2-p}\sum_{i=1}^{n} \Vert Z_{ii} \Vert _{p}^{p}\quad (0\leq p \leq 2). $$

(11)

Mao and Liu [17] showed the inequality

$$ \sum_{i\neq j} \Vert Z_{ij} \Vert _{p}^{p}\leq (n-1)2^{\frac{p}{2}} \sum_{i=1}^{n} \Vert Z_{ii} \Vert _{p}^{p}\quad (p> 0), $$

(12)

where for $0 < p\leq \frac{4}{3}$ and $p\geq 4$, this inequality improved inequalities (10) and (11). Lin and Fu [16], extended the above inequalities for sector matrices as follows:

$$ \sum_{i\neq j} \Vert Z_{ij} \Vert _{p}^{p}\leq (n-1)\sec ^{p}(\alpha ) \sum_{i=1}^{n} \Vert Z_{ii} \Vert _{p}^{p}\quad (p> 0), $$

(13)

in which $\alpha \in [0,\frac{\pi }{2} )$.

In the present paper, we establish some unitarily invariant norm inequalities for sector matrices involving the functions of class $\mathcal{C}$. For instance, we extend inequalities (2) and (6) to sector matrices and the class $\mathcal{C}$ (Theorem 4). Moreover, we improve inequalities (3) and (4) to sector matrices. Also, we prove inequality (13) for all unitarily invariant norm and function of the class $\mathcal{C}$.

2 Main result

In the following, we give some lemmas which are needed to prove our main statements.

Lemma 1

([9, p. 207])

Let$X,Y,Z\in {\mathcal{M}}_{n}$, andr, sbe positive real numbers with$\frac{1}{r}+\frac{1}{s}=1$. Then

$$ \Vert X \Vert _{w} \leq \Vert Y \Vert _{w}^{\frac{1}{r} } \Vert Z \Vert _{w}^{ \frac{1}{s}}, $$

where$w=(w_{1},w_{2},\ldots ,w_{n})$is a decreasing sequence of nonnegative real numbers if and only if

$$ \Vert X \Vert \leq \Vert Y \Vert ^{\frac{1}{r} } \Vert Z \Vert ^{ \frac{1}{s}} $$

for every unitarily invariant norm$\Vert \cdot \Vert$.

Lemma 2

([1, Theorem 3.2])

Suppose that$Z\in {\mathcal{M}}_{2n}$partitioned as in (1) such that$W(Z) \subseteq S_{\alpha }$for some$\alpha \in [0,\frac{\pi }{2})$. Then

$$ \prod_{m=1}^{k} s_{m}(Z_{ij}) \leq \prod_{l=1}^{k} \sec ( \alpha ) s_{m}^{\frac{1}{2}}\bigl({\mathcal{R}e}(Z_{ii})\bigr) s_{m}^{ \frac{1}{2}}\bigl({\mathcal{R}e}(Z_{jj})\bigr)\quad (i,j=1,2), $$

where$k=1,2,\ldots ,n$.

Lemma 3

([5, p. 73])

Let$Z\in {\mathcal{M}}_{n}$. Then

$$ \lambda _{j}\bigl({\mathcal{R}e}(Z)\bigr)\leq s_{j} ( Z )\quad (j=1,2, \ldots ,n). $$

Consequently, $\Vert {\mathcal{R}e}(Z)\Vert \leq \Vert Z\Vert$for every unitarily invariant norm$\Vert \cdot \Vert$on${\mathcal{M}}_{n}$.

In the sequel, we give some unitarily invariant norm inequalities for sector matrices regarding of special class $\mathcal{C}$. Furthermore, in some special cases those results reduce to previous ones, which were introduced by other authors.

Theorem 4

Let$Z\in {\mathcal{M}}_{2n}$partitioned as in (1) be a sector matrix and let$h \in \mathcal{C}$be submultiplicative and$\alpha \in [0,\frac{\pi }{2} )$. Ifrandsare positive real numbers with$\frac{1}{r}+\frac{1}{s}=1$, then

$$\begin{aligned} \bigl\Vert h \bigl( \vert Z_{ij} \vert ^{2} \bigr) \bigr\Vert &\leq \bigl\Vert h^{r} \bigl( \sec (\alpha ) { \mathcal{R}e} (Z_{11}) \bigr) \bigr\Vert ^{ \frac{1}{r} } \bigl\Vert h^{s} \bigl( \sec (\alpha ) {\mathcal{R}e} (Z_{22}) \bigr) \bigr\Vert ^{\frac{1}{s} } \\ &\leq \bigl\Vert h^{r} \bigl( \sec (\alpha ) \vert Z_{11} \vert \bigr) \bigr\Vert ^{\frac{1}{r} } \bigl\Vert h^{s} \bigl( \sec (\alpha ) \vert Z_{22} \vert \bigr) \bigr\Vert ^{\frac{1}{s} } \end{aligned}$$

for every unitarily invariant norm$\Vert \cdot \Vert$on${\mathcal{M}}_{n}$and$i,j=1,2$.

Proof

Assume that $w=(w_{1},w_{2},\ldots ,w_{n})$ is a decreasing sequence of nonnegative real numbers and $k=1,2,\ldots ,n$. Then Lemma 2 implies that

$$\begin{aligned} \prod_{m=1}^{k} s_{m} \bigl( \vert Z_{ij} \vert ^{2} \bigr) &= \Biggl(\prod _{m=1}^{k} s_{m}(Z_{ij}) \Biggr)^{2} \leq \Biggl( \prod_{m=1}^{k} \sec (\alpha ) s_{m}^{\frac{1}{2}}\bigl({ \mathcal{R}e}(Z_{ii}) \bigr) s_{m}^{\frac{1}{2}}\bigl({\mathcal{R}e}(Z_{jj}) \bigr) \Biggr)^{2} \\ &= \prod_{m=1}^{k} \sec ^{2}(\alpha ) s_{m}\bigl({\mathcal{R}e}(Z_{ii}) \bigr) s_{m} \bigl( {\mathcal{R}e}(Z_{jj}) \bigr), \end{aligned}$$

where $i,j=1,2$. Therefore

$$\begin{aligned} \prod_{m=1}^{k} s_{m} \bigl( h \bigl( \vert Z_{ij} \vert ^{2} \bigr) \bigr)&=\prod_{m=1}^{k} h \bigl( s_{m} \bigl( \vert Z_{ij} \vert ^{2} \bigr) \bigr)\quad (\text{since $ h $ is increasing}) \\ &\leq \prod_{m=1}^{k} h \bigl( \sec ^{2}(\alpha ) s_{m}\bigl({\mathcal{R}e}(Z_{ii}) \bigr) s_{m} \bigl( {\mathcal{R}e}(Z_{jj}) \bigr) \bigr) \\ & \quad ( \text{since $ f \in \mathcal{C}$}) \\ &\leq \prod_{m=1}^{k} h \bigl( \sec ( \alpha ) s_{m}\bigl({\mathcal{R}e}(Z_{ii})\bigr) \bigr) h \bigl( \sec (\alpha ) s_{m} \bigl( {\mathcal{R}e}(Z_{jj}) \bigr) \bigr) \\ & \quad ( \text{since $ h $ is submultiplicative}) \\ &= \prod_{m=1}^{k} s_{m} \bigl( h \bigl( \sec (\alpha ){\mathcal{R}e}(Z_{ii}) \bigr) \bigr) s_{m} \bigl( h \bigl( \sec (\alpha ) {\mathcal{R}e} ( Z_{jj} ) \bigr) \bigr). \end{aligned}$$

Since $w=(w_{1},w_{2},\ldots ,w_{n})$ is a decreasing sequence of nonnegative real numbers, it follows that

$$ \prod_{m=1}^{k} w_{m} s_{m} \bigl( h \bigl( \vert Z_{ij} \vert ^{2} \bigr) \bigr) \leq \prod_{m=1}^{k} w_{m} s_{m} \bigl( h \bigl( \sec (\alpha ){ \mathcal{R}e}(Z_{ii}) \bigr) \bigr) s_{m} \bigl( h \bigl( \sec (\alpha ) {\mathcal{R}e} ( Z_{jj} ) \bigr) \bigr), $$

(14)

where $i,j=1,2$. Since weak log majorization implies weak majorization, inequality (14) implies that

$$ \sum_{m=1}^{k} w_{m} s_{m} \bigl( h \bigl( \vert Z_{ij} \vert ^{2} \bigr) \bigr) \leq \sum_{m=1}^{k} w_{m} s_{m} \bigl( h \bigl( \sec (\alpha ){ \mathcal{R}e}(Z_{ii}) \bigr) \bigr) s_{m} \bigl( h \bigl( \sec (\alpha ) {\mathcal{R}e} ( Z_{jj} ) \bigr) \bigr), $$

(15)

where $i,j=1,2,\ldots$ . Now, by applying the previous inequality and Hölder’s inequality, we deduce that

$$\begin{aligned} & \bigl\Vert h \bigl( \vert Z_{12} \vert ^{2} \bigr) \bigr\Vert _{w} \\ &\quad = \sum_{m=1}^{n} w_{m} s_{m} \bigl( h \bigl( \vert Z_{12} \vert ^{2} \bigr) \bigr) \\ &\quad \leq \sum_{m=1}^{n} w_{m} s_{m} \bigl( h \bigl( \sec (\alpha ){ \mathcal{R}e}(Z_{11}) \bigr) \bigr) s_{m} \bigl( h \bigl( \sec ( \alpha ) {\mathcal{R}e} ( Z_{22} ) \bigr) \bigr) \\ & \qquad (\text{by inequality (15)}) \\ &\quad = \sum_{m=1}^{n} w_{m}^{\frac{1}{r}} s_{m} \bigl( h \bigl( \sec ( \alpha ){\mathcal{R}e}(Z_{11}) \bigr) \bigr) w_{m}^{\frac{1}{s}} s_{m} \bigl( h \bigl( \sec (\alpha ) {\mathcal{R}e} ( Z_{22} ) \bigr) \bigr) \\ &\quad \leq \Biggl( \sum_{m=1}^{n} w_{m} s_{m}^{r} \bigl( h \bigl( \sec ( \alpha ){\mathcal{R}e}(Z_{11}) \bigr) \bigr) \Biggr)^{\frac{1}{r}} \Biggl( \sum_{m=1}^{n} w_{m} s_{m}^{s} \bigl( h \bigl( \sec ( \alpha ) {\mathcal{R}e} ( Z_{22} ) \bigr) \bigr) \Biggr)^{\frac{1}{s}} \\ &\qquad (\text{by H\"{o}lder's inequality}) \\ &\quad = \Biggl( \sum_{m=1}^{n} w_{m} s_{m} \bigl( h^{r} \bigl( \sec ( \alpha ){\mathcal{R}e}(Z_{11}) \bigr) \bigr) \Biggr)^{\frac{1}{r}} \Biggl( \sum_{m=1}^{n} w_{m} s_{m} \bigl( h^{s} \bigl( \sec ( \alpha ) {\mathcal{R}e} ( Z_{22} ) \bigr) \bigr) \Biggr)^{\frac{1}{s}} \\ &\quad = \bigl\Vert h^{r} \bigl( \sec (\alpha ) {\mathcal{R}e} (Z_{11}) \bigr) \bigr\Vert _{w} ^{\frac{1}{r} } \bigl\Vert h^{s} \bigl( \sec (\alpha ) { \mathcal{R}e} (Z_{22}) \bigr) \bigr\Vert _{w } ^{\frac{1}{s} }. \end{aligned}$$

(16)

If we replace $w_{m}^{\frac{1}{r} }$ with $w_{m}^{\frac{1}{s} }$ in the third equality, then by a similar process we obtain

$$ \bigl\Vert h \bigl( \vert Z_{21} \vert ^{2} \bigr) \bigr\Vert _{w} \leq \bigl\Vert h^{r} \bigl( \sec (\alpha ) {\mathcal{R}e} (Z_{11}) \bigr) \bigr\Vert _{w} ^{ \frac{1}{r} } \bigl\Vert h^{s} \bigl( \sec (\alpha ) {\mathcal{R}e} (Z_{22}) \bigr) \bigr\Vert _{w} ^{\frac{1}{s} } $$

(17)

for all decreasing sequences $w=(w_{1},w_{2},\ldots ,w_{n})$ of nonnegative real numbers. It follows from Lemma 1 and inequalities (16) and (17) that

$$\begin{aligned} \bigl\Vert h \bigl( \vert Z_{ij} \vert ^{2} \bigr) \bigr\Vert &\leq \bigl\Vert h^{r} \bigl( \sec (\alpha ) { \mathcal{R}e} (Z_{11}) \bigr) \bigr\Vert ^{ \frac{1}{r} } \bigl\Vert h^{s} \bigl( \sec (\alpha ) {\mathcal{R}e} (Z_{22}) \bigr) \bigr\Vert ^{\frac{1}{s} } \\ &\leq \bigl\Vert h^{r} \bigl( \sec (\alpha ) \vert Z_{11} \vert \bigr) \bigr\Vert ^{\frac{1}{r} } \bigl\Vert h^{s} \bigl( \sec (\alpha ) \vert Z_{22} \vert \bigr) \bigr\Vert ^{\frac{1}{s} }\quad (i,j=1,2). \end{aligned}$$

 □

Remark 5

If $Z\in {\mathcal{M}}_{2n}$ is positive semidefinite, i.e. $W(Z) \subseteq S_{0}$, then Theorem 4 reduces to inequality (2). Applying Theorem 4 for $h(t)=t^{\frac{p}{2}}$ ($p>0$), we get inequality (6). Therefore Theorem 4 is an extension of inequality (2) and inequality (6).

Corollary 6

Suppose$Z\in {\mathcal{M}}_{2n}$partitioned as in (1) is accretive–dissipative and$h \in \mathcal{C}$is submultiplicative. Ifrandsare positive real numbers with$\frac{1}{r}+\frac{1}{s}=1$, then

$$ \bigl\Vert h \bigl( \vert Z_{ij} \vert ^{2} \bigr) \bigr\Vert \leq \bigl\Vert h^{r} \bigl( \sqrt{2} {\mathcal{R}e} (Z_{11}) \bigr) \bigr\Vert ^{\frac{1}{r} } \bigl\Vert h^{s} \bigl( \sqrt{2} {\mathcal{R}e}(Z_{22}) \bigr) \bigr\Vert ^{ \frac{1}{s} }\quad ( i,j=1,2), $$

where$\Vert \cdot \Vert$is a unitarily invariant norm.

Proof

Since Z is accretive–dissipative, i.e. $W(e^{\frac{-i\pi }{4}}Z) \subseteq S_{\frac{\pi }{4}}$ and $\sec (\frac{\pi }{4})=\sqrt{2}$, by applying Theorem 4, we get the statement. □

Corollary 7

([2, Theorem 4.2])

Let$Z\in {\mathcal{M}}_{2n}$partitioned as in (1) such that$W(Z) \subseteq S_{\alpha }$for some$\alpha \in [0,\frac{\pi }{2})$. Then

$$\begin{aligned} \bigl\Vert \vert Z_{12}\vert ^{p} \bigr\Vert ^{2} & \leq \sec ^{2p}(\alpha ) \bigl\Vert Z_{11}^{p} \bigr\Vert \bigl\Vert Z_{22}^{p} \bigr\Vert \\ &\leq \sec ^{2p}(\alpha ) \bigl\Vert \vert Z_{11}\vert ^{p}\bigr\Vert \bigl\Vert \vert Z_{22}\vert ^{p}\bigr\Vert \quad (p>0) \end{aligned}$$

for every unitarily invariant norm.

Proof

Applying Theorem 4 for $r=2$, $s=2$ and $h(t)=t^{\frac{p}{2}}$ ($p>0$), we get

$$\begin{aligned} \bigl\Vert \vert Z_{12}\vert ^{p} \bigr\Vert ^{2} & \leq \sec ^{2p}(\alpha ) \bigl\Vert {\mathcal{R}e}(Z_{11})^{p} \bigr\Vert \bigl\Vert {\mathcal{R}e}(Z_{22})^{p} \bigr\Vert \\ &\leq \sec ^{2p}(\alpha ) \bigl\Vert Z_{11}^{p} \bigr\Vert \bigl\Vert Z_{22}^{p} \bigr\Vert \\ &\leq \sec ^{2p}(\alpha ) \bigl\Vert \vert Z_{11}\vert ^{p}\bigr\Vert \bigl\Vert \vert Z_{22}\vert ^{p}\bigr\Vert \quad (p>0). \end{aligned}$$

 □

Corollary 8

([22, Theorem 3.2])

Let$Z\in {\mathcal{M}}_{2n}$partitioned as in (1) such that$W(Z) \subseteq S_{\alpha }$for some$\alpha \in [0,\frac{\pi }{2})$. Then

$$\begin{aligned} \max \bigl\lbrace \Vert Z_{12} \Vert ^{2} , \Vert Z_{21} \Vert ^{2} \bigr\rbrace &\leq \sec ^{2}(\alpha ) \bigl\Vert {\mathcal{R}e}(Z_{11}) \bigr\Vert \bigl\Vert {\mathcal{R}e}(Z_{22}) \bigr\Vert \\ &\leq \sec ^{2}(\alpha ) \Vert Z_{11} \Vert \Vert Z_{22} \Vert \end{aligned}$$

(18)

for every unitarily invariant norm.

Proof

Applying Theorem 4 for $r=2$, $s=2$ and $h(t)=\sqrt{t}$, we get

$$ \bigl\Vert \vert Z_{12} \vert \bigr\Vert = \Vert Z_{12} \Vert \leq \bigl\Vert \sec ( \alpha ) {\mathcal{R}e}(Z_{11}) \bigr\Vert ^{\frac{1}{2}} \bigl\Vert \sec ( \alpha ) {\mathcal{R}e}(Z_{22}) \bigr\Vert ^{\frac{1}{2}}. $$

Therefore

$$ \Vert Z_{12} \Vert ^{2} \leq \sec ^{2}( \alpha ) \bigl\Vert {\mathcal{R}e}(Z_{11}) \bigr\Vert \bigl\Vert {\mathcal{R}e}(Z_{22}) \bigr\Vert \leq \sec ^{2}(\alpha ) \Vert Z_{11} \Vert \Vert Z_{22} \Vert . $$

Similarly, we have

$$\begin{aligned} \Vert Z_{21} \Vert ^{2} &\leq \sec ^{2}( \alpha ) \bigl\Vert {\mathcal{R}e}(Z_{11}) \bigr\Vert \bigl\Vert {\mathcal{R}e}(Z_{22}) \bigr\Vert \\ &\leq \sec ^{2} (\alpha ) \Vert Z_{11} \Vert \Vert Z_{22} \Vert . \end{aligned}$$

The above inequalities imply the expected result. □

Corollary 9

([22])

Let$Z\in {\mathcal{M}}_{2n}$partitioned as in (1) such that$W(Z) \subseteq S_{\alpha }$for some$\alpha \in [0,\frac{\pi }{2})$. Then, for any unitarily invariant norm, we have

$$\begin{aligned} 2 \Vert Z_{12} \Vert \Vert Z_{21} \Vert &\leq \Vert Z_{12} \Vert ^{2} + \Vert Z_{21} \Vert ^{2} \\ &\leq 2\sec ^{2}(\alpha ) \Vert Z_{11} \Vert \Vert Z_{22} \Vert . \end{aligned}$$

Proof

By using the arithmetic–geometric mean inequality and inequality (18), we have

$$\begin{aligned} 2 \Vert Z_{12} \Vert \Vert Z_{21} \Vert &\leq \Vert Z_{12} \Vert ^{2} + \Vert Z_{21} \Vert ^{2} \\ &\leq 2\max \bigl\lbrace \Vert Z_{12} \Vert ^{2} , \Vert Z_{21} \Vert ^{2} \bigr\rbrace \\ &\leq 2\sec ^{2}(\alpha ) \Vert Z_{11} \Vert \Vert Z_{22} \Vert . \end{aligned}$$

 □

Remark 10

Assume that h is a nonnegative increasing function on $[0, \infty )$. Since $s_{m} ( \vert Z_{ij} \vert ^{2} )=s_{m} ( \vert Z_{ij}^{*} \vert ^{2} )$ for $m=1,2,\ldots ,n$ and $i,j=1,2$, we have

$$ h \bigl( s_{m} \bigl( \vert Z_{ij} \vert ^{2} \bigr) \bigr)= s_{m} \bigl( h \bigl( \vert Z_{ij} \vert ^{2} \bigr) \bigr)=s_{m} \bigl( h \bigl( \bigl\vert Z_{ij}^{*} \bigr\vert ^{2} \bigr) \bigr)=h \bigl( s_{m} \bigl( \bigl\vert Z_{ij}^{*} \bigr\vert ^{2} \bigr) \bigr) $$

for $m=1,2,\ldots ,n$ and $i,j=1,2$. Therefore $\Vert h ( \vert Z_{ij} \vert ^{2} ) \Vert =\Vert h ( \vert Z_{ij}^{*} \vert ^{2} ) \Vert$.

Theorem 11

Suppose that$Z\in {\mathcal{M}}_{2n}$partitioned as in (1) is a sector matrix and$h \in \mathcal{C}$is submultiplicative convex. Ifrandsare positive real numbers with$\frac{1}{r}+\frac{1}{s}=1$, then

$$\begin{aligned} \bigl\Vert h \bigl( \vert Z_{12} \vert ^{2} \bigr) + h \bigl( \bigl\vert Z_{21}^{*} \bigr\vert ^{2} \bigr) \bigr\Vert &\leq \bigl\Vert h^{r}\bigl( \sqrt{2}\sec (\alpha ) { \mathcal{R}e} (Z_{11}) \bigr) \bigr\Vert ^{\frac{1}{r} } \bigl\Vert h^{s}\bigl( \sqrt{2} \sec (\alpha ) { \mathcal{R}e} (Z_{22}) \bigr) \bigr\Vert ^{\frac{1}{s} } \\ &\leq \bigl\Vert h^{r}\bigl( \sqrt{2}\sec (\alpha ) \vert Z_{11} \vert \bigr) \bigr\Vert ^{ \frac{1}{r} } \bigl\Vert h^{s}\bigl( \sqrt{2}\sec (\alpha ) \vert Z_{22} \vert \bigr) \bigr\Vert ^{\frac{1}{s} }, \end{aligned}$$

where$\alpha \in [0,\frac{\pi }{2} )$.

Proof

Applying the triangle inequality, Remark 10 and Theorem 4, we have

$$\begin{aligned} \bigl\Vert h \bigl( \vert Z_{12} \vert ^{2} \bigr)+h \bigl( \bigl\vert Z_{21}^{*} \bigr\vert ^{2} \bigr) \bigr\Vert &\leq \bigl\Vert h \bigl( \vert Z_{12} \vert ^{2} \bigr) \bigr\Vert + \bigl\Vert h \bigl( \bigl\vert Z_{21}^{*} \bigr\vert ^{2} \bigr) \bigr\Vert \\ &= \bigl\Vert h \bigl( \vert Z_{12} \vert ^{2} \bigr) \bigr\Vert + \bigl\Vert h \bigl( \vert Z_{21} \vert ^{2} \bigr) \bigr\Vert \\ &\leq 2 \bigl\Vert h^{r} \bigl(\sec (\alpha ) {\mathcal{R}e} (Z_{11}) \bigr) \bigr\Vert ^{\frac{1}{r} } \bigl\Vert h^{s} \bigl( \sec (\alpha ){ \mathcal{R}e} (Z_{22}) \bigr) \bigr\Vert ^{\frac{1}{s} }. \end{aligned}$$

It is well known that, if h is a convex function, then $h(\lambda Z)\geq \lambda h(Z)$ for $Z\in {\mathcal{M}}_{n}$ and $\lambda \geq 1$. Since $\sec (\alpha )\geq 1$ ($\alpha \in [0,\frac{\pi }{2} )$), we have

$$\begin{aligned} \bigl\Vert h \bigl( \vert Z_{12} \vert ^{2} \bigr)+h \bigl( \bigl\vert Z_{21}^{*} \bigr\vert ^{2} \bigr) \bigr\Vert &\leq \bigl\Vert h^{r}\bigl( \sqrt{2}\sec (\alpha ) { \mathcal{R}e} (Z_{11}) \bigr) \bigr\Vert ^{\frac{1}{r} } \bigl\Vert h^{s}\bigl( \sqrt{2} \sec (\alpha ) { \mathcal{R}e} (Z_{22}) \bigr) \bigr\Vert ^{\frac{1}{s} } \\ &\leq \bigl\Vert h^{r}\bigl( \sqrt{2}\sec (\alpha ) \vert Z_{11} \vert \bigr) \bigr\Vert ^{ \frac{1}{r} } \bigl\Vert h^{s}\bigl( \sqrt{2}\sec (\alpha ) \vert Z_{22} \vert \bigr) \bigr\Vert ^{\frac{1}{s} }. \end{aligned}$$

 □

Remark 12

Note that, if $Z\in {\mathcal{M}}_{2n}$ is accretive–dissipative, i.e. $W(e^{\frac{-i\pi }{4}}Z) \subseteq S_{\frac{\pi }{4}}$, then Theorem 11 reduces to inequality (3).

Theorem 13

Assume that$Z\in {\mathcal{M}}_{2n}$partitioned as in (1) is a sector matrix and$h \in \mathcal{C}$is submultiplicative concave. Ifrandsare positive real numbers with$\frac{1}{r}+\frac{1}{s}=1$, then

$$\begin{aligned} \bigl\Vert h \bigl( \vert Z_{12} \vert ^{2} \bigr) + h \bigl( \bigl\vert Z_{21}^{*} \bigr\vert ^{2} \bigr) \bigr\Vert &\leq 2\sec ^{2}(\alpha ) \bigl\Vert h^{r} \bigl( { \mathcal{R}e} (Z_{11}) \bigr) \bigr\Vert ^{\frac{1}{r} } \bigl\Vert h^{s} \bigl( {\mathcal{R}e} (Z_{22}) \bigr) \bigr\Vert ^{\frac{1}{s} } \\ &\leq 2\sec ^{2}(\alpha ) \bigl\Vert h^{r} \bigl( \vert Z_{11} \vert \bigr) \bigr\Vert ^{\frac{1}{r} } \bigl\Vert h^{s} \bigl( \vert Z_{22} \vert \bigr) \bigr\Vert ^{\frac{1}{s} } \end{aligned}$$

for every unitarily invariant norm$\Vert \cdot \Vert$and$\alpha \in [0,\frac{\pi }{2} )$.

Proof

Applying the triangle inequality, Remark 10 and Theorem 4, we have

$$\begin{aligned} \bigl\Vert h \bigl( \vert Z_{12} \vert ^{2} \bigr)+h \bigl( \bigl\vert Z_{21}^{*} \bigr\vert ^{2} \bigr) \bigr\Vert &\leq \bigl\Vert h \bigl( \vert Z_{12} \vert ^{2} \bigr) \bigr\Vert + \bigl\Vert h \bigl( \bigl\vert Z_{21}^{*} \bigr\vert ^{2} \bigr) \bigr\Vert \\ &= \bigl\Vert h \bigl( \vert Z_{12} \vert ^{2} \bigr) \bigr\Vert + \bigl\Vert h \bigl( \vert Z_{21} \vert ^{2} \bigr) \bigr\Vert \\ &\leq 2 \bigl\Vert h^{r} \bigl(\sec (\alpha ) {\mathcal{R}e} (Z_{11}) \bigr) \bigr\Vert ^{\frac{1}{r} } \bigl\Vert h^{s} \bigl( \sec (\alpha ){ \mathcal{R}e} (Z_{22}) \bigr) \bigr\Vert ^{\frac{1}{s} }. \end{aligned}$$

Since h is concave, it follows that $h(\lambda Z)\leq \lambda h(Z)$ for $Z\in {\mathcal{M}}_{n}$ and $\lambda \geq 1$. Since $\sec (\alpha )\geq 1$ for $\alpha \in [0,\frac{\pi }{2} )$,

$$\begin{aligned} \bigl\Vert h \bigl( \vert Z_{12} \vert ^{2} \bigr)+h \bigl( \bigl\vert Z_{21}^{*} \bigr\vert ^{2} \bigr) \bigr\Vert &\leq 2\sec ^{2}(\alpha ) \bigl\Vert h^{r} \bigl( { \mathcal{R}e} (Z_{11}) \bigr) \bigr\Vert ^{\frac{1}{r} } \bigl\Vert h^{s} \bigl( {\mathcal{R}e} (Z_{22}) \bigr) \bigr\Vert ^{\frac{1}{s} } \\ &\leq 2\sec ^{2}(\alpha ) \bigl\Vert h^{r} \bigl( \vert Z_{11} \vert \bigr) \bigr\Vert ^{\frac{1}{r} } \bigl\Vert h^{s} \bigl( \vert Z_{22} \vert \bigr) \bigr\Vert ^{\frac{1}{s} }. \end{aligned}$$

 □

Remark 14

If $Z\in {\mathcal{M}}_{2n}$ is accretive–dissipative, i.e. $W(e^{\frac{-i\pi }{4}}Z) \subseteq S_{\frac{\pi }{4}}$, then Theorem 13 reduces to inequality (4).

Theorem 15

Assume that$Z\in {\mathcal{M}}_{2n}$partitioned as in (1) is a sector matrix, $h \in \mathcal{C}$is submultiplicative and$\alpha \in [0,\frac{\pi }{2} )$. Ifpis positive real number, then

$$ \bigl\Vert h \bigl( \vert Z_{12} \vert ^{2} \bigr) \bigr\Vert ^{p} + \bigl\Vert h \bigl( \vert Z_{21} \vert ^{2} \bigr) \bigr\Vert ^{p} \leq 2 \bigl\Vert h^{2} \bigl( \sec (\alpha ) \vert Z_{11} \vert \bigr) \bigr\Vert ^{\frac{p}{2} } \bigl\Vert h^{2} \bigl( \sec (\alpha ) \vert Z_{22} \vert \bigr) \bigr\Vert ^{ \frac{p}{2} } $$

for every unitarily invariant norm$\Vert \cdot \Vert$. In particular, we have

$$ \bigl\Vert h \bigl( \vert Z_{12} \vert ^{2} \bigr) \bigr\Vert _{p}^{p} + \bigl\Vert h \bigl( \vert Z_{21} \vert ^{2} \bigr) \bigr\Vert _{p}^{p} \leq 2 \bigl\Vert h^{2} \bigl( \sec (\alpha ) \vert Z_{11} \vert \bigr) \bigr\Vert _{p} ^{ \frac{p}{2} } \bigl\Vert h^{2} \bigl( \sec ( \alpha ) \vert Z_{22} \vert \bigr) \bigr\Vert _{p} ^{\frac{p}{2} }. $$

Proof

Theorem 4 for $r=s=2$, implies that

$$ \bigl\Vert h \bigl( \vert Z_{ij} \vert ^{2} \bigr) \bigr\Vert \leq \bigl\Vert h^{2} \bigl( \sec (\alpha ) \vert Z_{11} \vert \bigr) \bigr\Vert ^{\frac{1}{2} } \bigl\Vert h^{2} \bigl( \sec (\alpha ) \vert Z_{22} \vert \bigr) \bigr\Vert ^{ \frac{1}{2} }\quad (i,j=1,2). $$

(19)

By taking the power p of both sides of inequality (19), we have

$$ \bigl\Vert h \bigl( \vert Z_{ij} \vert ^{2} \bigr) \bigr\Vert ^{p} \leq \bigl\Vert h^{2} \bigl( \sec ( \alpha ) \vert Z_{11} \vert \bigr) \bigr\Vert ^{\frac{p}{2} } \bigl\Vert h^{2} \bigl( \sec (\alpha ) \vert Z_{22} \vert \bigr) \bigr\Vert ^{ \frac{p}{2} } \quad ( i,j=1,2). $$

Therefore, we have

$$ \bigl\Vert h \bigl( \vert Z_{12} \vert ^{2} \bigr) \bigr\Vert ^{p} + \bigl\Vert h \bigl( \vert Z_{21} \vert ^{2} \bigr) \bigr\Vert ^{p} \leq 2 \bigl\Vert h^{2} \bigl( \sec (\alpha ) \vert Z_{11} \vert \bigr) \bigr\Vert ^{\frac{p}{2} } \bigl\Vert h^{2} \bigl( \sec (\alpha ) \vert Z_{22} \vert \bigr) \bigr\Vert ^{ \frac{p}{2} }. $$

 □

Corollary 16

([16, Theorem 2.8])

Let$Z\in {\mathcal{M}}_{2n}$be partitioned as in (1) such that$W(Z) \subseteq S_{\alpha }$for some$\alpha \in [0,\frac{\pi }{2})$. Then, for any unitarily invariant norm, we have

$$ \Vert Z_{12} \Vert ^{p} + \Vert Z_{21} \Vert ^{p} \leq 2\sec ^{p}( \alpha ) \Vert Z_{11} \Vert ^{\frac{p}{2}} \Vert Z_{22} \Vert ^{ \frac{p}{2}}\quad ( p>0). $$

In particular, we have

$$ \Vert Z_{12} \Vert _{p} ^{p} + \Vert Z_{21} \Vert _{p}^{p} \leq 2 \sec ^{p}(\alpha ) \Vert Z_{11} \Vert _{p}^{\frac{p}{2}} \Vert Z_{22} \Vert _{p}^{\frac{p}{2}}\quad ( p>0). $$

Proof

Applying Theorem 15, for $h(t)=\sqrt{t}$, we have

$$ \Vert Z_{12} \Vert ^{p} + \Vert Z_{21} \Vert ^{p} \leq 2\sec ^{p}( \alpha ) \Vert Z_{11} \Vert ^{\frac{p}{2}} \Vert Z_{22} \Vert ^{ \frac{p}{2}} \quad ( p>0). $$

By showing the particular case, by using the Schatten p-norm, we have the statement. □

In the sequel, we extend our results to $n\times n$ block matrices as introduced in (9).

Theorem 17

Suppose thatZis a sector matrix represented as in (9), $h \in \mathcal{C}$is submultiplicative and$\alpha \in [0,\frac{\pi }{2} )$. Ifpis positive real number, then

$$ \sum_{i\neq j} \bigl\Vert h \bigl( \vert Z_{ij} \vert ^{2} \bigr) \bigr\Vert ^{p} \leq (n-1)\sum_{i=1}^{n} \bigl\Vert h^{2} \bigl( \sec (\alpha ) \vert Z_{ii} \vert \bigr) \bigr\Vert ^{p} $$

(20)

for every unitarily invariant norm$\Vert \cdot \Vert$. In particular, we have

$$ \sum_{i\neq j} \bigl\Vert h \bigl( \vert Z_{ij} \vert ^{2} \bigr) \bigr\Vert _{p}^{p} \leq (n-1)\sum_{i=1}^{n} \bigl\Vert h^{2} \bigl( \sec (\alpha ) \vert Z_{ii} \vert \bigr) \bigr\Vert _{p} ^{p}. $$

Proof

Since Z is a sector matrix, so every principal submatrix of Z is also a sector matrix, it follows that $\left(\begin{array}{cc}{Z}_{ii}& Z_{ij}\\ T_{ji}& Z_{jj}\end{array}\right)$ is a sector matrix. Now, applying Theorem 15 for $\left(\begin{array}{cc}{Z}_{ii}& Z_{ij}\\ Z_{ji}& Z_{jj}\end{array}\right)$, we get

$$ \bigl\Vert h \bigl( \vert Z_{ij} \vert ^{2} \bigr) \bigr\Vert ^{p} + \bigl\Vert h \bigl( \vert Z_{ji} \vert ^{2} \bigr) \bigr\Vert ^{p} \leq 2 \bigl\Vert h^{2} \bigl( \sec (\alpha ) \vert Z_{ii} \vert \bigr) \bigr\Vert ^{\frac{p}{2} } \bigl\Vert h^{2} \bigl( \sec (\alpha ) \vert Z_{jj} \vert \bigr) \bigr\Vert ^{ \frac{p}{2} } $$

for $i\neq j$. By using the arithmetic–geometric mean inequality, we have

$$ \bigl\Vert h \bigl( \vert Z_{ij} \vert ^{2} \bigr) \bigr\Vert ^{p} + \bigl\Vert h \bigl( \vert Z_{ji} \vert ^{2} \bigr) \bigr\Vert ^{p} \leq \bigl\Vert h^{2} \bigl( \sec (\alpha ) \vert Z_{ii} \vert \bigr) \bigr\Vert ^{p}+ \bigl\Vert h^{2} \bigl( \sec ( \alpha ) \vert Z_{jj} \vert \bigr) \bigr\Vert ^{p} $$

for $i\neq j$. Adding the previous inequalities for $i, j = 1, 2,\ldots , n$, we get

$$ \sum_{i\neq j} \bigl\Vert h \bigl( \vert Z_{ij} \vert ^{2} \bigr) \bigr\Vert ^{p} \leq (n-1)\sum_{i=1}^{n} \bigl\Vert h^{2} \bigl( \sec (\alpha ) \vert Z_{ii} \vert \bigr) \bigr\Vert ^{p}. $$

 □

Corollary 18

([16, Theorem 2.9])

LetZbe a sector matrix as represented in (9) and$\alpha \in [0,\frac{\pi }{2} )$. Then

$$ \sum_{i\neq j} \Vert Z_{ij} \Vert ^{p} \leq (n-1)\sec ^{p}(\alpha ) \sum _{i=1}^{n} \Vert Z_{ii} \Vert ^{p} \quad (p > 0), $$

(21)

for any unitarily invariant norm. In particular, we have

$$ \sum_{i\neq j} \Vert Z_{ij} \Vert _{p}^{p}\leq (n-1)\sec ^{p}(\alpha ) \sum _{i=1}^{n} \Vert Z_{ii} \Vert _{p}^{p}\quad (p> 0). $$

Proof

Applying Theorem 17, for $h(t)=\sqrt{t}$, we have

$$ \sum_{i\neq j} \Vert Z_{ij} \Vert ^{p} \leq (n-1)\sec (\alpha ) \sum_{i=1}^{n} \Vert Z_{ii} \Vert ^{p}\quad (p > 0). $$

For the particular case, we take the Schatten p-norm. □