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Norm inequalities for submultiplicative functions involving contraction sector \(2 \times 2\) block matrices

Abstract

In this article, we show unitarily invariant norm inequalities for sector \(2\times 2\) block matrices which extend and refine some recent results of Bourahli, Hirzallah, and Kittaneh (Positivity, 2020, https://doi.org/10.1007/s11117-020-00770-w).

Introduction

Let \(\mathbb{M}_{n}\) be a set of all \(n\times n\) complex matrices. A matrix \(A\in \mathbb{M}_{n}\) is said to be positive semidefinite if \(x^{*}Ax\geq 0\) for all \(x\in \mathbb{C}^{n}\). If the eigenvalues \(\lambda _{1}(A), \dots , \lambda _{n}(A)\) of A are all real, we arrange them in nonincreasing order \(\lambda _{1}(A)\ge \cdots \ge \lambda _{n}(A)\). Singular values of A are the eigenvalues of \(|A|\) and are arranged in nonincreasing order \(s_{1}(A)\ge \cdots \ge s_{n}(A)\). For \(A \in \mathbb{M}_{n}\), we denote by \(|A|=(A^{*}A)^{\frac{1}{2}}\), \(A^{*}\), \(\|A\|\), and \(\|A\|_{\infty }=s_{1}(A)\) the absolute value, the conjugate transpose, the unitarily invariant norm, and the operator norm, respectively. We say A is a contraction if \(\|A\|_{\infty }\leq 1\). By convention, the \(n \times n\) identity matrix is denoted by \(I_{n}\). \(\|A\|\) and \(\|A\|_{\infty }=s_{1}(A)\) are unitarily invariant, i.e., \(\|UAV\|=\|A\|\) for all unitary matrices U, V. For \(A, B\in \mathbb{M}_{n}\), the weak majorization relation \(s(A)\prec _{w} s(B)\) means

$$ \sum_{j=1}^{k}s_{j}(A) \leq \sum _{j=1}^{k}s_{j}(B),\quad k=1,2, \ldots , n. $$

For \(A\in \mathbb{M}_{n}\), recall the Cartesian (or Toeplitz) decomposition (see, e.g., [2, p. 6] and [3, p. 7])

$$ A =\operatorname{Re}A+i \operatorname{Im}A, $$

where

$$ \operatorname{Re}A:=\frac{1}{2}\bigl(A+A^{*}\bigr),\qquad \operatorname{Im}A:= \frac{1}{2i}\bigl(A-A^{*}\bigr). $$

The Cartesian decomposition of a matrix is unique. There are many interesting properties for such a decomposition. A celebrated result due to Fan and Hoffman (see, e.g., [2, p.73]) states that

$$\begin{aligned} \lambda _{j}(\operatorname{Re}A) \leq & s_{j}(A),\quad j=1, \ldots , n. \end{aligned}$$
(1)

The numerical range of \(A\in \mathbb{M}_{n}\) is defined by

$$ W(A) = \bigl\{ x^{*}Ax\mid x\in \mathbb{C}^{n}, x^{*}x=1\bigr\} , $$

which is a compact convex set (see, e.g., [4, Chap. 1]). For \(\alpha \in [0, \pi /2)\), a sector on the complex plane is

$$ S_{\alpha } =\bigl\{ z\in \mathbb{C}\mid \operatorname{Re}z \geq 0, \vert \operatorname{Im}z \vert \leq (\operatorname{Re}z)\tan \alpha \bigr\} . $$

A sector matrix \(A\in \mathbb{M}_{n}\) is a matrix whose numerical range is contained in \(S_{\alpha }\) for some \(\alpha \in [0, \pi /2)\). It is clear that if \(A\in \mathbb{M}_{n}\) is a sector matrix, then ReA is positive semidefinite. The interested readers can refer to [510], and [4] for recent results on sector matrices. If \(W(A)\) is contained in the first quadrant of the complex plane, then ReA and ImA are positive semidefinite. We call such a matrix A accretive-dissipative. Note that if A is accretive-dissipative, then \(W(e^{-\frac{i\pi }{4}}A)\subseteq S_{\frac{\pi }{4}}\). Recently this class of matrices has been studied by researchers partly due to the fact that it contains the class of positive semidefinite matrices (see, e.g., [1, 1117]).

Next we introduce a special class of functions. Let \(\mathcal{C}\) be the class of all nonnegative increasing functions f on \([0, \infty )\) preserving the weak-log majorization, i.e., for two nonincreasing sequences of nonnegative real numbers \((x_{1}, x_{2},\ldots , x_{n})\) and \((y_{1}, y_{2}, \ldots , y_{n})\), \(\prod_{j=1}^{k} x_{j} \leq \prod_{j=1}^{k} y_{j}\) for \(k=1,\ldots , n\) implies \(\prod_{j=1}^{k} f(x_{j}) \leq \prod_{j=1}^{k} f(y_{j})\) for \(k=1,\ldots , n\). There are many other properties on this class of functions; see [12, 18]. A nonnegative function \(f\in \mathcal{C}\) on the interval \([0, \infty )\) is said to be submultiplicative if \(f(ab)\leq f(a)f(b)\) whenever \(a, b\in [0, \infty )\). Recently, some unitarily invariant norm inequalities for submultiplicative functions of accretive-dissipative matrices have been shown in [12] and [13].

Bourahli et al. [1, Lemma 3.4] showed that if A= ( A 11 A 12 A 21 A 22 ) M 2 n is a positive semidefinite contraction and s, t are positive real numbers such that \(\frac{1}{s}+\frac{1}{t}=1\), then

$$ \bigl\Vert f\bigl( \vert A_{12} \vert ^{2}\bigr) \bigr\Vert \leq \bigl\Vert f^{s} \bigl(A_{11}^{\frac{1}{2}} \bigr) \bigr\Vert ^{\frac{1}{s}} \bigl\Vert f^{t} \bigl(A_{22}^{\frac{1}{2}} \bigr) \bigr\Vert ^{\frac{1}{t}}, $$
(2)

where f is an increasing submultiplicative function on \([0, \infty )\) with \(f(0)=0\). Moreover, if \(A\in \mathbb{M}_{2n}\) is just positive semidefinite (not necessarily contraction matrices), then they presented a result related to (2) in [1, Remark 3.5] as follows:

$$ \bigl\Vert f\bigl( \vert A_{12} \vert ^{2}\bigr) \bigr\Vert \leq f \bigl( \bigl\Vert A_{11}^{\frac{1}{2}} \bigr\Vert _{\infty } \bigl\Vert A_{22}^{\frac{1}{2}} \bigr\Vert _{\infty } \bigr) \bigl\Vert f^{s} \bigl(A_{11}^{\frac{1}{2}} \bigr) \bigr\Vert ^{ \frac{1}{s}} \bigl\Vert f^{t} \bigl(A_{22}^{\frac{1}{2}} \bigr) \bigr\Vert ^{ \frac{1}{t}}. $$
(3)

Unitarily invariant norms for submultiplicative functions

In [12] and [13], some unitarily invariant norms for accretive-dissipative matrices involving a special class of functions have been shown. In this section, we present inequalities for sector block matrices involving the class of function.

Lemma 2.1

([19, p. 280])

Let A, X, B be \(m\times p\), \(p\times q\), \(q\times n\) matrices, respectively. Then

$$ s_{i}(AXB)\leq s_{1}(A)s_{j}(X)s_{1}(B), \quad i\leq \min \{m, p, q, n \}. $$

Lemma 2.2

([8, Theorem 2.1])

Let \(A\in \mathbb{M}_{n}\) be \(n\times n\) such that \(W(A)\subseteq S_{\alpha }\) for some \(\alpha \in [0, \pi /2)\). Then there exist an invertible matrix X and a unitary and diagonal matrix \(Z=\operatorname{diag}(e^{i\theta _{1}},\ldots , e^{i\theta _{n}})\) with all \(|\theta _{j}|\leq \alpha \) such that \(A=XZX^{*}\). Moreover, such a matrix Z is unique up to permutation.

Lemma 2.3

([8, Corollary 2.3 (ii)])

Let \(A\in \mathbb{M}_{n}\) be such that \(W(A)\subseteq S_{\alpha }\) for some \(\alpha \in [0, \pi /2)\), and let \(A=XZX^{*}\) be a sectoral decomposition of A, where X is invertible and Z is unitary and diagonal. Then

$$\begin{aligned} RR^{*}\leq \sec (\alpha ) \bigl(R(\operatorname{Re}Z)R^{*} \bigr)=\sec (\alpha ) \bigl( \operatorname{Re}\bigl(RZR^{*}\bigr)\bigr) \end{aligned}$$

for every matrix \(R\in \mathbb{M}_{n}\).

We are ready to present our main result of this section.

Theorem 2.4

Let \(f\in \mathcal{C}\) be an increasing submultiplicative function on \([0, \infty )\) and \(A\in \mathbb{M}_{2n}\) be a contraction matrix partitioned as

$$ A= \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix},$$
(4)

with \(W(A)\subseteq S_{\alpha }\) for some \(\alpha \in [0, \pi /2)\). Then, for all \(r, s, t>0\) with \(\frac{1}{s}+\frac{1}{t}=1\) and all unitarily invariant norms,

$$ \bigl\Vert f\bigl( \vert A_{12} \vert ^{2r}\bigr) \bigr\Vert \leq \bigl\Vert f^{s}\bigl(\bigl( \sec ^{2}(\alpha ) \vert A_{11} \vert \bigr)^{r/2}\bigr) \bigr\Vert ^{1/s} \bigl\Vert f^{t}\bigl(\bigl(\sec ^{2}(\alpha ) \vert A_{22} \vert \bigr)^{r/2}\bigr) \bigr\Vert ^{1/t} $$
(5)

and

$$ \bigl\Vert f \bigl( \vert A_{21} \vert ^{2r} \bigr) \bigr\Vert \le \bigl\Vert f^{s}\bigl(\bigl( \sec ^{2}( \alpha ) \vert A_{11} \vert \bigr)^{r/2}\bigr) \bigr\Vert ^{1/s} \bigl\Vert f^{t}\bigl(\bigl(\sec ^{2}(\alpha ) \vert A_{22} \vert \bigr)^{r/2}\bigr) \bigr\Vert ^{1/t}. $$
(6)

Proof

Note that A is a sector matrix with \(W(A)\subseteq S_{\alpha }\). By Lemma 2.2, we have \(A=XZX^{*}\), where X is invertible and Z is unitary and diagonal. We partition X as ( X 1 X 2 ) , \(X_{1},X_{2}\in \mathbb{M}_{n \times 2n}\). Thus, \(\operatorname{Re}A_{11}=X_{1}(\operatorname{Re}Z)X_{1}^{*}\), \(\operatorname{Re}A_{22}=X_{2}(\operatorname{Re}Z)X_{2}^{*}\), and \(A_{12}=X_{1}ZX_{2}^{*}\). Consider the Cartesian decomposition

$$\begin{aligned} A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix}=R+iS= \begin{pmatrix} R_{11} & R_{12} \\ R_{21} & R_{22} \end{pmatrix}+i \begin{pmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{pmatrix}, \end{aligned}$$

where R is positive semidefinite and S is Hermitian. Since A is a contraction matrix,

$$\begin{aligned} AA^{*}=R^{2}+S^{2}+i(SR-RS)\leq I \end{aligned}$$
(7)

and

$$\begin{aligned} A^{*}A=R^{2}+S^{2}+i(RS-SR)\leq I. \end{aligned}$$
(8)

Adding (7) and (8), we get

$$\begin{aligned} 2\bigl(R^{2}+S^{2}\bigr) \leq 2I. \end{aligned}$$

Thus, R and S are also contraction matrices, which implies that both \(\operatorname{Re}A_{11}=R_{11}\) and \(\operatorname{Re}A_{22}=R_{22}\) are positive semidefinite contractions.

Now

$$\begin{aligned} s_{\ell }\bigl( \vert A_{12} \vert ^{r}\bigr) =&s_{\ell }^{r}\bigl( \vert A_{12} \vert \bigr) =s_{\ell }^{r}(A_{12})= s_{\ell }^{r}\bigl(X_{1}ZX_{2}^{*} \bigr) \\ \leq & s_{1}^{r}(X_{1})s_{\ell }^{r} \bigl(ZX_{2}^{*}\bigr) \quad \text{(by Lemma 2.1)} \\ =& \lambda _{1}^{\frac{r}{2}}\bigl(X_{1}^{*}X_{1} \bigr)\lambda _{\ell }^{ \frac{r}{2}}\bigl(X_{2}Z^{*}ZX_{2}^{*} \bigr) \\ = &\lambda _{1}^{\frac{r}{2}}\bigl(X_{1}X_{1}^{*} \bigr)\lambda _{\ell }^{ \frac{r}{2}}\bigl(X_{2}X_{2}^{*} \bigr) \\ \le &\lambda _{1}^{\frac{r}{2}}\bigl(\sec (\alpha )X_{1}( \operatorname{Re}Z)X_{1}^{*}\bigr)\lambda _{\ell }^{\frac{r}{2}}\bigl(\sec ( \alpha )X_{2}( \operatorname{Re}Z)X_{2}^{*}\bigr) \quad \text{(by Lemma 2.3)} \\ =&\lambda _{1}^{\frac{r}{2}}\bigl(\sec (\alpha ) \operatorname{Re}A_{11}\bigr) \lambda _{\ell }^{\frac{r}{2}} \bigl(\sec (\alpha )\operatorname{Re}A_{22}\bigr) \end{aligned}$$
(9)
$$\begin{aligned} \le & \sec ^{\frac{r}{2}}(\alpha ) \lambda _{\ell }^{\frac{r}{2}} \bigl( \sec (\alpha )\operatorname{Re}A_{22}\bigr)\quad ( \text{since } \operatorname{Re}A_{11}\text{ is a contraction}) \\ \le & \sec ^{r}(\alpha ) s_{\ell }^{\frac{r}{2}} \bigl( \vert A_{22} \vert \bigr)\quad ( \text{by (1)}) \end{aligned}$$
(10)

for \(l=1,2, \ldots , n\).

Since \(\operatorname{Re}A_{22}\) is also a contraction, it follows from (1) and (9) that

$$\begin{aligned} s_{\ell }\bigl( \vert A_{12} \vert ^{r}\bigr) \leq & \sec ^{r}(\alpha ) s_{\ell }^{\frac{r}{2}} \bigl( \vert A_{11} \vert \bigr) \end{aligned}$$
(11)

for \(l=1,2, \ldots , n\).

Multiplying inequalities (10) and (11) by each other implies that

$$\begin{aligned} s_{\ell }\bigl( \vert A_{12} \vert ^{2r}\bigr) \leq & \sec ^{2r}(\alpha ) s_{\ell }^{ \frac{r}{2}} \bigl( \vert A_{11} \vert \bigr)s_{\ell }^{\frac{r}{2}} \bigl( \vert A_{22} \vert \bigr) \end{aligned}$$
(12)

for \(l=1,2, \ldots , n\).

So,

$$\begin{aligned} s_{\ell }\bigl(f\bigl( \vert A_{12} \vert ^{2r}\bigr)\bigr) =&f\bigl(s_{\ell }\bigl( \vert A_{12} \vert ^{2r}\bigr)\bigr) \\ \leq &f\bigl(\sec ^{2r}(\alpha ) s_{\ell }^{\frac{r}{2}} \bigl( \vert A_{11} \vert \bigr)s_{\ell }^{ \frac{r}{2}} \bigl( \vert A_{22} \vert \bigr)\bigr) \quad \text{(by (12))} \\ = &f\bigl(\sec ^{r}(\alpha ) s_{\ell }^{\frac{r}{2}} \bigl( \vert A_{11} \vert \bigr)\bigr)f\bigl(\sec ^{r}( \alpha ) s_{\ell }^{\frac{r}{2}}\bigl( \vert A_{22} \vert \bigr)\bigr)\\ & (\text{since }f\text{ is submultiplicative}) \end{aligned}$$
(13)

for \(l=1,2, \ldots , n\). Let \(\alpha =(\alpha _{1},\alpha _{2}, \ldots , \alpha _{n})\) be a decreasing sequence of nonnegative real numbers. The α-norm of a matrix \(B\in \mathbb{M}_{n}\) is defined by

$$ \Vert B \Vert _{\alpha } = \sum_{\ell =1}^{n} \alpha _{\ell }s_{\ell }(B). $$

The α-norms are unitarily invariant [4, p. 204].

Actually, inequality (13) means that

$$ \prod_{\ell =1}^{k}\alpha _{\ell }s_{\ell } \bigl(f\bigl( \vert A_{12} \vert ^{2r}\bigr)\bigr) \leq \prod_{\ell =1}^{k}\alpha _{\ell }s_{\ell } \bigl(f\bigl(\bigl(\sec ^{2}(\alpha ) \vert A_{11} \vert \bigr)^{ \frac{r}{2}}\bigr)\bigr)s_{\ell }\bigl(f\bigl(\bigl(\sec ^{2}(\alpha ) \vert A_{22} \vert \bigr)^{\frac{r}{2}} \bigr)\bigr) $$

for \(k=1,2, \ldots , n\), which implies that

$$\begin{aligned} \begin{aligned} \sum _{\ell =1}^{k}\alpha _{\ell }s_{\ell } \bigl(f\bigl( \vert A_{12} \vert ^{2r}\bigr)\bigr) &\leq \sum_{\ell =1}^{k}\alpha _{\ell }s_{\ell } \bigl(f\bigl(\bigl(\sec ^{2}(\alpha ) \vert A_{11} \vert \bigr)^{ \frac{r}{2}}\bigr)\bigr)s_{\ell }\bigl(f\bigl(\bigl(\sec ^{2}(\alpha ) \vert A_{22} \vert \bigr)^{\frac{r}{2}} \bigr)\bigr) \end{aligned} \end{aligned}$$

for \(k=1,2, \ldots , n\). Thus,

$$\begin{aligned}& \bigl\Vert f\bigl( \vert A_{12} \vert ^{2r}\bigr) \bigr\Vert _{\alpha } \\& \quad = \sum _{\ell =1}^{n}\alpha _{\ell }s_{\ell } \bigl(f\bigl( \vert A_{12} \vert ^{2r}\bigr)\bigr) \\& \quad \leq \sum_{\ell =1}^{n}\alpha _{\ell }s_{\ell }\bigl(f\bigl(\bigl(\sec ^{2}(\alpha ) \vert A_{11} \vert \bigr)^{r/2}\bigr)\bigr)s_{\ell } \bigl(f\bigl(\bigl(\sec ^{2}(\alpha ) \vert A_{22} \vert \bigr)^{r/2}\bigr)\bigr) \\& \quad = \sum_{\ell =1}^{n}\alpha _{\ell }^{1/s}s_{\ell }\bigl(f\bigl(\bigl(\sec ^{2}( \alpha ) \vert A_{11} \vert \bigr)^{r/2} \bigr)\bigr)\alpha _{\ell }^{1/t}s_{\ell }\bigl(f\bigl( \bigl(\sec ^{2}( \alpha ) \vert A_{22} \vert \bigr)^{r/2}\bigr)\bigr) \\& \quad \leq \Biggl(\sum_{\ell =1}^{n}\alpha _{\ell }s_{\ell }^{s}\bigl(f\bigl(\bigl(\sec ^{2}( \alpha ) \vert A_{11} \vert \bigr)^{r/2} \bigr)\bigr) \Biggr)^{1/s} \Biggl(\sum_{\ell =1}^{m} \alpha _{\ell }s_{\ell }^{t}\bigl(f\bigl(\bigl(\sec ^{2}(\alpha ) \vert A_{22} \vert \bigr)^{r/2} \bigr)\bigr) \Biggr)^{1/t} \\& \quad \bigl(\text{by H\"{o}lder's inequality} \bigr) \\& \quad = \Biggl(\sum_{\ell =1}^{n}\alpha _{\ell }s_{\ell }\bigl(f^{s}\bigl(\bigl(\sec ^{2}( \alpha ) \vert A_{11} \vert \bigr)^{r/2} \bigr)\bigr) \Biggr)^{1/s} \Biggl(\sum_{\ell =1}^{m} \alpha _{\ell }s_{\ell }\bigl(f^{t}\bigl(\bigl(\sec ^{2}(\alpha ) \vert A_{22} \vert \bigr)^{r/2} \bigr)\bigr) \Biggr)^{1/t} \\& \quad = \bigl\Vert f^{s}\bigl(\bigl(\sec ^{2}(\alpha ) \vert A_{11} \vert \bigr)^{r/2}\bigr) \bigr\Vert _{\alpha }^{1/s} \bigl\Vert f^{t}\bigl(\bigl(\sec ^{2}(\alpha ) \vert A_{22} \vert \bigr)^{r/2} \bigr) \bigr\Vert _{\alpha }^{1/t} \end{aligned}$$

for all decreasing sequences \(\alpha =(\alpha _{1},\ldots , \alpha _{n})\) of nonnegative real numbers. It follows from the above inequality that

$$ \bigl\Vert f\bigl( \vert A_{12} \vert ^{2r}\bigr) \bigr\Vert \leq \bigl\Vert f^{s}\bigl(\bigl(\sec ^{2}( \alpha ) \vert A_{11} \vert \bigr)^{r/2}\bigr) \bigr\Vert ^{1/s} \bigl\Vert f^{t}\bigl(\bigl(\sec ^{2}( \alpha ) \vert A_{22} \vert \bigr)^{r/2}\bigr) \bigr\Vert ^{1/t}. $$

The inequality for \(A_{21}\) is similarly proved. □

Remark 1

In particular, when A is a positive semidefinite contraction (\(\alpha =0\)) and \(r=1\), Theorem 2.4 gives

$$ \bigl\Vert f\bigl( \vert A_{12} \vert ^{2}\bigr) \bigr\Vert \le \bigl\Vert f^{s} \bigl(A_{11}^{1/2}\bigr) \bigr\Vert ^{1/s} \bigl\Vert f^{t}\bigl(A_{22}^{1/2}\bigr) \bigr\Vert ^{1/t}, $$
(14)

which is due to Bourahli et al. [1, Lemma 3.4]. Thus, our result (5) is a generalization of (14).

Remark 2

If A is just a general sector matrix with \(W(A)\subseteq S_{\alpha }\) for \(\alpha \in [0, \frac{\pi }{2})\) (not a contraction matrix), then we have the following result: Let \(f\in \mathcal{C}\) be an increasing submultiplicative function on \([0, \infty )\) and A= ( A 11 A 12 A 21 A 22 ) M 2 n be a sector matrix with \(W(A)\subseteq S_{\alpha }\) for some \(\alpha \in [0, \pi /2)\). Then, for all \(r, s, t>0\) with \(\frac{1}{s}+\frac{1}{t}=1\) and all unitarily invariant norms,

$$\begin{aligned} \bigl\Vert f\bigl( \vert A_{12} \vert ^{2r}\bigr) \bigr\Vert = &f \bigl(\sec ^{2r}(\alpha ) \Vert A_{11} \Vert _{ \infty }^{\frac{r}{2}} \Vert A_{22} \Vert _{\infty }^{\frac{r}{2}} \bigr) \bigl\Vert f^{s}\bigl( \vert A_{11} \vert ^{ \frac{r}{2}}\bigr) \bigr\Vert ^{\frac{1}{s}} \bigl\Vert f^{t}\bigl( \vert A_{22} \vert ^{\frac{r}{2}}\bigr) \bigr\Vert ^{ \frac{1}{t}}. \end{aligned}$$
(15)

By (9), (10), and Lemma 2.1, we have

$$\begin{aligned} s_{\ell }\bigl( \vert A_{12} \vert ^{r}\bigr) =&s_{\ell }^{r}\bigl( \vert A_{12} \vert \bigr) =s_{\ell }^{r}(A_{12})= s_{\ell }^{r}\bigl(X_{1}ZX_{2}^{*} \bigr) \\ \le & \sec ^{r}(\alpha ) \Vert A_{11} \Vert _{\infty }^{\frac{r}{2}}s_{\ell }^{ \frac{r}{2}}\bigl( \vert A_{22} \vert \bigr) \end{aligned}$$
(16)

and

$$\begin{aligned} s_{\ell }\bigl( \vert A_{12} \vert ^{r}\bigr) \leq & \sec ^{r}(\alpha ) \Vert A_{22} \Vert _{\infty }^{ \frac{r}{2}}s_{\ell }^{\frac{r}{2}} \bigl( \vert A_{11} \vert \bigr) \end{aligned}$$
(17)

for \(l=1,2, \ldots , n\).

Multiplying inequalities (16) and (17) by each other implies that

$$\begin{aligned} s_{\ell }\bigl( \vert A_{12} \vert ^{2r}\bigr) \leq & \sec ^{2r}(\alpha ) \Vert A_{11} \Vert _{\infty }^{ \frac{r}{2}} \Vert A_{22} \Vert _{\infty }^{\frac{r}{2}}s_{\ell }^{\frac{r}{2}}\bigl( \vert A_{11} \vert \bigr)s_{\ell }^{\frac{r}{2}}\bigl( \vert A_{22} \vert \bigr) \end{aligned}$$
(18)

for \(l=1,2, \ldots , n\).

So,

$$\begin{aligned} s_{\ell }\bigl(f\bigl( \vert A_{12} \vert ^{2r}\bigr)\bigr) =&f\bigl(s_{\ell }\bigl( \vert A_{12} \vert ^{2r}\bigr)\bigr) \\ \leq &f(\sec ^{2r}(\alpha ) \Vert A_{11} \Vert _{\infty }^{\frac{r}{2}} \Vert A_{22} \Vert _{\infty }^{\frac{r}{2}}s_{\ell }^{\frac{r}{2}}\bigl( \vert A_{11} \vert \bigr)s_{\ell }^{ \frac{r}{2}}\bigl( \vert A_{22} \vert \bigr) \quad \text{(by (18))} \\ = &f \bigl(\sec ^{2r}(\alpha ) \Vert A_{11} \Vert _{\infty }^{\frac{r}{2}} \Vert A_{22} \Vert _{\infty }^{\frac{r}{2}} \bigr)f \bigl(s_{\ell }^{\frac{r}{2}} \bigl( \vert A_{11} \vert \bigr)\bigr)f\bigl(s_{\ell }^{\frac{r}{2}} \bigl( \vert A_{22} \vert \bigr) \bigr) \end{aligned}$$
(19)

for \(l=1,2, \ldots , n\). Based on (19), we can obtain the desired result by a proof similar to that given for inequality (13). Therefore, when \(\alpha =0\) and \(r=1\), our result (15) is (2).

Theorem 2.5

Let \(f\in \mathcal{C}\) be an increasing submultiplicative function on \([0, \infty )\) and \(A\in \mathbb{M}_{2n}\) be a contraction matrix partitioned as

$$ A= \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix},$$

with \(W(A)\subseteq S_{\alpha }\) for some \(\alpha \in [0, \pi /2)\). Then, for all \(r, s, t>0\) with \(\frac{1}{s}+\frac{1}{t}=1\) and all unitarily invariant norms,

$$ \begin{aligned}[b] &\bigl\Vert f\bigl( \vert A_{12} \vert ^{2r}\bigr) \bigr\Vert + \bigl\Vert f \bigl( \vert A_{21} \vert ^{2r} \bigr) \bigr\Vert \\ &\quad \leq 2 \bigl\Vert f^{s}\bigl(\bigl( \sec ^{2}(\alpha ) \vert A_{11} \vert \bigr)^{r/2}\bigr) \bigr\Vert ^{1/s} \bigl\Vert f^{t}\bigl(\bigl(\sec ^{2}( \alpha ) \vert A_{22} \vert \bigr)^{r/2}\bigr) \bigr\Vert ^{1/t}. \end{aligned} $$
(20)

Proof

By Theorem 2.4, we can have the desired result. □

Remark 3

When \(f(t)=t\), \(r=1\), and \(\alpha =\frac{\pi }{4}\), result (20) becomes

$$ \bigl\Vert \vert A_{12} \vert ^{2} \bigr\Vert + \bigl\Vert \vert A_{21} \vert ^{2} \bigr\Vert \leq 2 \bigl\Vert f^{s}\bigl(\bigl(\sec ^{2}(\alpha ) \vert A_{11} \vert \bigr)^{r/2}\bigr) \bigr\Vert ^{1/s} \bigl\Vert f^{t}\bigl(\bigl( \sec ^{2}( \alpha ) \vert A_{22} \vert \bigr)^{r/2}\bigr) \bigr\Vert ^{1/t}. $$

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Acknowledgements

The author wants to express her sincere thanks to the referee for his or her valuable remarks and suggestions, which made this paper more readable.

Funding

Zhou’s work was supported by the High-level Talents Program of Natural Science Foundation of Hainan Province, China (2019RC193) and the Scientific Research Project of Colleges and Universities of Hainan Province, China (Hnky2020-31).

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Correspondence to Xiaoying Zhou.

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Zhou, X. Norm inequalities for submultiplicative functions involving contraction sector \(2 \times 2\) block matrices. J Inequal Appl 2020, 247 (2020). https://doi.org/10.1186/s13660-020-02514-6

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MSC

  • 15A18
  • 15A45

Keywords

  • Contraction sector block matrices
  • Unitarily invariant norms
  • Submultiplicative functions