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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=(Z11Z12Z21Z22) 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.

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, 1115, 17, 1922] 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 (XZZY)M2n, 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 \(\mathcal{C}\).

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 (ZiiZijTjiZjj) is a sector matrix. Now, applying Theorem 15 for (ZiiZijZjiZjj), 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. □

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Afraz, D., Lashkaripour, R. & Bakherad, M. Norm inequalities involving a special class of functions for sector matrices. J Inequal Appl 2020, 122 (2020). https://doi.org/10.1186/s13660-020-02383-z

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MSC

  • 47A63
  • 15A60

Keywords

  • Unitarily invariant norm
  • Accrative–dissipative matrix
  • Numerical range
  • Sector matrix