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 accretiveâ€“dissipative.

Another important class of matrices, which is related to the class of accretiveâ€“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 accretiveâ€“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, 11â€“15, 17, 19â€“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 $\left(\begin{array}{cc}X& Z\\ {Z}^{âˆ—}& Y\end{array}\right)âˆˆ{\mathcal{M}}_{2n}$, 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 accretiveâ€“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 accretiveâ€“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 accretiveâ€“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 $$i,j=1,2,\ldots$$â€‰. Now, by applying the previous inequality and HÃ¶lderâ€™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}

â€ƒâ–¡

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 accretiveâ€“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 accretiveâ€“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.â€ƒâ–¡

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}

â€ƒâ–¡

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.â€ƒâ–¡

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 arithmeticâ€“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}

â€ƒâ–¡

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}

â€ƒâ–¡

Remark 12

Note that, if $$Z\in {\mathcal{M}}_{2n}$$ is accretiveâ€“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}

â€ƒâ–¡

Remark 14

If $$Z\in {\mathcal{M}}_{2n}$$ is accretiveâ€“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

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

â€ƒâ–¡

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.â€ƒâ–¡

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 arithmeticâ€“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

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

â€ƒâ–¡

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.â€ƒâ–¡

References

1. Alakhrass, M.: A note on sectorial matrices. Linear Multilinear Algebra (to appear). https://doi.org/10.1080/03081087.2019.1575332

2. Alakhrass, M., Sababheh, M.: Lieb functions and sectorial matrices. Linear Algebra Appl. 586, 308â€“324 (2020)

3. Arlinski, Y.M., Popov, A.B.: On sectorial matrices. Linear Algebra Appl. 370, 133â€“146 (2003)

4. Bakherad, M., Lashkaripour, R., Hajmohamadi, M.: Extensions of interpolation between the arithmeticâ€“geometric mean inequality for matrices. J. Inequal. Appl. 2017, 209 (2017)

5. Bhatia, R.: Matrix Analysis. Springer, New York (1997)

6. Drury, S., Lin, M.: Singular value inequalities for matrices with numerical ranges in a sector. Oper. Matrices 8(4), 1143â€“1148 (2014)

7. Fu, X., He, C.: On some Fischer-type determinantal inequalities for accretiveâ€“dissipative matrices. J. Inequal. Appl. 2013, 316 (2013)

8. Gumus, I.H., Hirzallah, O., Kittaneh, F.: Norm inequalities involving accretiveâ€“dissipative $$2 \times 2$$ block matrices. Linear Algebra Appl. 528, 76â€“93 (2017)

9. Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)

10. Kittaneh, F., Sakkijha, M.: Inequalities for accretiveâ€“dissipative matrices. Linear Multilinear Algebra 67(5), 1037â€“1042 (2019)

11. Li, C.K., Sze, N.: Determinantal and eigenvalue inequalities for matrices with numerical ranges in a sector. J. Math. Anal. Appl. 410(1), 487â€“491 (2014)

12. Lin, M.: Fischer type determinantal inequalities for accretiveâ€“dissipative matrices. Linear Algebra Appl. 438, 2808â€“2812 (2013)

13. Lin, M.: A note on the growth factor in Gaussian elimination for accretiveâ€“dissipative matrices. Calcolo 51, 363â€“366 (2014)

14. Lin, M.: Some inequalities for sector matrices. Oper. Matrices 10(4), 915â€“921 (2016)

15. Lin, M., Zhou, D.: Norm inequalities for accretiveâ€“dissipative operator matrices. J. Math. Anal. Appl. 407, 436â€“442 (2013)

16. Lin, S., Fu, X.: On some inequalities for sector matrices. Linear Multilinear Algebra (to appear). https://doi.org/10.1080/03081087.2019.1600466

17. Mao, Y., Liu, X.: On some inequalities for accretiveâ€“dissipative matrices. Linear Multilinear Algebra (to appear). https://doi.org/10.1080/03081087.2019.1635566

18. Shebrawi, K., Bakherad, M.: Generalizations of the Aluthge transform of operators. Filomat 32(18), 6465â€“6474 (2018)

19. Yang, C., Lu, F.: Some generalizations of inequalities for sector matrices. J. Inequal. Appl. 2018, 183 (2018)

20. Yang, J.: Some determinantal inequalities for accretiveâ€“dissipative matrices. J. Inequal. Appl. 2013, 512 (2013)

21. Zhang, D., Hou, L., Ma, L.: Properties of matrices with numerical ranges in a sector. Bull. Iran. Math. Soc. 43, 1699â€“1707 (2017)

22. Zhang, F.: A matrix decomposition and its applications. Linear Multilinear Algebra 63(10), 2033â€“2042 (2015)

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We thank the anonymous referees for reading the paper carefully and providing thoughtful comments.

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Correspondence to Rahmatollah Lashkaripour.

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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