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Some inequalities related to \(2\times 2\) block sector partial transpose matrices

Abstract

In this article, two inequalities related to \(2\times 2\) block sector partial transpose matrices are proved, and we also present a unitarily invariant norm inequality for the Hua matrix which is sharper than an existing result.

Introduction

We denote by \(\mathbb{M}_{n}\) the set of \(n\times n\) complex matrices. \(\mathbb{M}_{n}(\mathbb{M}_{k})\) is the set of \(n\times n\) block matrices with each block in \(\mathbb{M}_{k}\). The \(n\times n\) identity matrix is denoted by \(I_{n}\). We use \(\|\cdot \|\) for an arbitrary unitarily invariant norm. A positive semidefinite matrix A will be expressed as \(A\geq 0\). Likewise, we write \(A>0\) to refer that A is a positive definite matrix. The singular values of A, denoted by \(s_{1}(A), s_{2}(A),\ldots, s_{n}(A)\), are the eigenvalues of the positive semidefinite matrix \(|A|=(A^{*}A)^{1/2}\), arranged in decreasing order and repeated according to multiplicity as \(s_{1}(A)\geq s_{2}(A)\geq \cdots \geq s_{n}(A)\). When A is Hermitian, we enumerate eigenvalues of A in nonincreasing order \(\sigma _{1}(A)\geq \sigma _{2}(A)\geq \cdots \geq \sigma _{n}(A)\). Recall that \(C\in \mathbb{M}_{m\times n}\) is (strictly) contractive if (\(I_{n}>C^{*}C\)) \({ I_{n} \geq C^{*}C}\). The geometric mean of two positive definite matrices \(A, B\in \mathbb{M}_{n}\), denoted by \(A\sharp B\), is the positive definite solution of the Riccati equation \(XB^{-1}X=A\) and has the explicit expression \(A\sharp B=A^{\frac{1}{2}}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^{ \frac{1}{2}}A^{\frac{1}{2}}\). More details on the matrix geometric mean can be found in [2, Chap. 4].

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

$$\begin{aligned} W(A)=\bigl\{ x^{*}Ax|x\in \mathbb{C}^{n}, x^{*}x=1 \bigr\} . \end{aligned}$$

For basic properties of numerical range, see [5]. Also, we define a sector on the complex plane

$$\begin{aligned} S_{\alpha }=\bigl\{ z\in \mathbb{C }|\mathfrak{R}z\geq 0, \vert \mathfrak{I}z \vert \leq (\mathfrak{R}z)\tan (\alpha )\bigr\} ,\quad \alpha \in\biggl[0, \frac{\pi }{2}\biggr). \end{aligned}$$

Actually, the class of matrices T with \(W(T)\subseteq S_{\alpha }\) and the class of T with positive definite real part (i.e. accretive matrices) are both called sector matrices. Sector matrices have been the subject of a number of recent papers [3, 8, 14].

A matrix \(H=(H_{ij})_{i,j=1}^{n}\in \mathbb{M}_{n}(\mathbb{M}_{k})\) is said to be positive partial transpose (i.e. PPT) if H is positive semidefinite and its partial transpose \(H^{\tau }=(H_{ji})_{j,i=1}^{n}\) is also positive semidefinite. Inspired by PPT, Kuai [6] defined a new conception called sectorial partial transpose (i.e. SPT). That is, if \(W(A)\subseteq S_{\alpha }\) for \(A=(A_{ij})_{i,j=1}^{n}\in \mathbb{M}_{n}(\mathbb{M}_{k})\), then \(W(A^{\tau })\subseteq S_{\alpha }\). Thus, it is natural to extend the results for PPT matrices to SPT matrices.

Hiroshima [4, Theorem 1] proved the following result.

Theorem 1.1

LetH=(AXXB)M2(Mn)be PPT. Then

$$\begin{aligned} \Vert H \Vert \leq \Vert A+B \Vert . \end{aligned}$$
(1)

As the application of Theorem 1.1, Lin and Hiroshima [10, Theorem 3.3] presented a relation between the norm of diagonal blocks of the Hua matrix, e.g., [12] and the norm of its off diagonal blocks.

Theorem 1.2

If the Hua matrix is given by

$$\begin{aligned} H: = \begin{pmatrix} (I-A^{*}A)^{-1} & (I-B^{*}A)^{-1} \\ (I-A^{*}B)^{-1} & (I-B^{*}B)^{-1} \end{pmatrix}, \end{aligned}$$

where\(A,B \in \mathbb{M}_{m\times n}\)are strictly contractive, then

$$\begin{aligned} 2 \bigl\Vert \bigl(I-A^{*}B\bigr)^{-1} \bigr\Vert \leq \bigl\Vert \bigl(I-A^{*}A\bigr)^{-1}+ \bigl(I-B^{*}B\bigr)^{-1} \bigr\Vert \end{aligned}$$
(2)

for any unitarily invariant norm.

Actually, it was only recently observed that H is PPT; see [1].

Lin [7] obtained a singular value inequality for PPT matrices related to a linear map.

Theorem 1.3

Let\(A, B, X\in \mathbb{M}_{n}\). If

$$ M= \begin{pmatrix} A & X \\ X^{*} & B \end{pmatrix} $$

is PPT, then for the linear map\(\varPhi: C\rightarrow C+\operatorname{Tr}(C)I\),

$$\begin{aligned} s_{j}\bigl(\varPhi (X)\bigr) \leq s_{j}\bigl(\varPhi (A \sharp B)\bigr),\quad j=1,\ldots, n. \end{aligned}$$

In this paper, we extend Theorem 1.1 and Theorem 1.3 to SPT matrices and show a stronger inequality than (2).

Main results

We start with some lemmas. The first three lemmas are quite standard in matrix analysis.

Lemma 2.1

([13, p. 63])

If\(H\in \mathbb{M}_{n}\), then

$$\begin{aligned} \sigma _{j}(\operatorname {Re}H) \leq s_{j} (H),\quad j=1, \ldots,n. \end{aligned}$$
(3)

Lemma 2.2

([11, Theorem 1])

LetH=(AXXB)Mm+nbe positive semidefinite with\(A\in \mathbb{M}_{m}, B\in \mathbb{M}_{n}\). Then

$$\begin{aligned} 2s_{j}(X) \leq s_{j}(H),\quad j=1,\ldots, \min \{m, n\}. \end{aligned}$$
(4)

Lemma 2.3

([2, p.106])

Let\(A, B\in \mathbb{M}_{n}\)be positive definite matrices. Then, for all\(X\in M_{n}\),

$$\begin{aligned} X^{*}(A\sharp B)X \leq \bigl(X^{*}AX\bigr) \sharp \bigl(X^{*}BX\bigr). \end{aligned}$$
(5)

The next lemma is due to Zhang [14, Lemma 3.1].

Lemma 2.4

Let\(A\in \mathbb{M}_{n}\)have\(W(A)\subseteq S_{\alpha }\)for some\(\alpha \in [0, \frac{\pi }{2})\). Then

$$\begin{aligned} \Vert A \Vert \leq \sec (\alpha ) \Vert \operatorname {Re}A \Vert \end{aligned}$$
(6)

for any unitarily invariant norm.

The following result about geometric mean has been proved by Lin and Sun [9].

Lemma 2.5

Let\(A, B\in \mathbb{M}_{n}\)be matrices with positive semidefinite real part. Then

$$\begin{aligned} (\operatorname {Re}A)\sharp (\operatorname {Re}B) \leq \operatorname {Re}(A\sharp B). \end{aligned}$$
(7)

Now we are ready to present our results. The first theorem is an extension of Theorem 1.1.

Theorem 2.6

Let\(H_{11}, H_{12}, H_{21}, H_{22}\in \mathbb{M}_{n}\). IfH=(H11H12H21H22)is SPT, then

$$\begin{aligned} \Vert H \Vert \leq \sec (\alpha ) \Vert H_{11}+H_{22} \Vert \end{aligned}$$

for any unitarily invariant norm.

Proof

Since H is a sector partial transpose matrix, then we know that

$$\operatorname {Re}H= \begin{pmatrix} \frac{ H_{11}+H_{11}^{*}}{2} & \frac{ H_{12}+H_{21}^{*}}{2} \\ \frac{ H_{21}+H_{12}^{*}}{2} &\frac{ H_{22}+H_{22}^{*}}{2} \end{pmatrix} $$

is PPT.

So by (6) we have

$$\begin{aligned} \Vert H \Vert &\leq \sec (\alpha ) \Vert \operatorname {Re}H \Vert \\ &\leq \sec (\alpha ) \Vert \operatorname {Re}H_{11}+ \operatorname {Re}H_{22} \Vert \quad\bigl(\text{by (1)}\bigr) \\ &\leq \sec (\alpha ) \Vert H_{11}+H_{22} \Vert . \end{aligned}$$

 □

Remark 2.7

When \(H_{12}=H_{21}^{*}\) and \(\alpha =0\), then H is PPT in Theorem 2.6. Thus, our result is Hiroshima’s inequality (1).

Next will give a stronger inequality than Theorem 1.2.

Theorem 2.8

Let the Hua matrix be given by

$$\begin{aligned} H:= \begin{pmatrix} (I-A^{*}A)^{-1} & (I-B^{*}A)^{-1} \\ (I-A^{*}B)^{-1} & (I-B^{*}B)^{-1} \end{pmatrix}, \end{aligned}$$

where\(A,B \in \mathbb{M}_{m\times n}\)are strictly contractive. Then

$$\begin{aligned} \bigl\Vert \bigl(I-A^{*}B\bigr)^{-1} \bigr\Vert \leq \bigl\Vert \bigl(I-A^{*}A\bigr)^{-1}\sharp \bigl(I-B^{*}B\bigr)^{-1} \bigr\Vert \end{aligned}$$

for any unitarily invariant norm.

Proof

Since H is PPT, then

$$\begin{aligned} \begin{pmatrix} (I-A^{*}A)^{-1} & (I-B^{*}A)^{-1} \\ (I-A^{*}B)^{-1} & (I-B^{*}B)^{-1} \end{pmatrix},\qquad \begin{pmatrix} (I-A^{*}A)^{-1} & (I-A^{*}B)^{-1} \\ (I-B^{*}A)^{-1} & (I-B^{*}B)^{-1} \end{pmatrix} \end{aligned}$$

are both positive semidefinite matrices.

Hence,

$$\begin{aligned} \bigl(I-B^{*}B\bigr)^{-1} \geq \bigl(I-A^{*}B\bigr)^{-1}\bigl(I-A^{*}A\bigr) \bigl(I-B^{*}A\bigr)^{-1} \end{aligned}$$

and

$$\begin{aligned} \bigl(I-B^{*}B\bigr)^{-1} \geq \bigl(I-B^{*}A\bigr)^{-1} \bigl(I-A^{*}A\bigr) \bigl(I-A^{*}B\bigr)^{-1}. \end{aligned}$$
(8)

Clearly, by unitary similarity transformation,

$$ \begin{pmatrix} (I-B^{*}B)^{-1} & (I-A^{*}B)^{-1} \\ (I-B^{*}A)^{-1} & (I-A^{*}A)^{-1} \end{pmatrix} $$

is also positive semidefinite.

Therefore,

$$\begin{aligned} \bigl(I-A^{*}A\bigr)^{-1} \geq \bigl(I-B^{*}A\bigr)^{-1} \bigl(I-B^{*}B\bigr) \bigl(I-A^{*}B\bigr)^{-1}. \end{aligned}$$
(9)

Thus,

$$\begin{aligned} &\bigl(I-B^{*}B\bigr)^{-1}\sharp \bigl(I-A^{*}A \bigr)^{-1} \\ &\qquad{}- \bigl(I-B^{*}A\bigr)^{-1}\bigl(\bigl(I-A^{*}A \bigr)^{-1}\sharp \bigl(I-B^{*}B\bigr)^{-1} \bigr)^{-1}\bigl(I-A^{*}B\bigr)^{-1} \\ &\quad\geq \bigl(I-B^{*}B\bigr)^{-1}\sharp \bigl(I-A^{*}A \bigr)^{-1} \\ &\qquad{}-\bigl(\bigl(I-B^{*}A\bigr)^{-1}\bigl(I-A^{*}A \bigr) \bigl(I-A^{*}B\bigr)^{-1}\bigr)\sharp \bigl( \bigl(I-B^{*}A\bigr)^{-1}\bigl(I-B^{*}B\bigr) \bigl(I-A^{*}B\bigr)^{-1}\bigr) \\ & \qquad\bigl(\text{by (5) and monotonicity}\bigr) \\ &\quad\geq \bigl(I-B^{*}B\bigr)^{-1}\sharp \bigl(I-A^{*}A \bigr)^{-1}-\bigl(I-B^{*}B\bigr)^{-1}\sharp \bigl(I-A^{*}A\bigr)^{-1} \quad \bigl(\text{by (8) and (9)}\bigr) \\ &\quad=0. \end{aligned}$$

In a similar way, we can prove

$$\begin{aligned} &\bigl(I-B^{*}B\bigr)^{-1}\sharp \bigl(I-A^{*}A \bigr)^{-1} \\ &\quad{}- \bigl(I-A^{*}B\bigr)^{-1}\bigl(\bigl(I-A^{*}A \bigr)^{-1}\sharp \bigl(I-B^{*}B\bigr)^{-1} \bigr)^{-1}\bigl(I-B^{*}A\bigr)^{-1} \geq 0. \end{aligned}$$

So

$$ K:= \begin{pmatrix} (I-A^{*}A)^{-1}\sharp (I-B^{*}B)^{-1} & (I-A^{*}B)^{-1} \\ (I-B^{*}A)^{-1} & (I-B^{*}B)^{-1}\sharp (I-A^{*}A)^{-1} \end{pmatrix} $$

is PPT.

Therefore,

$$\begin{aligned} 2 \bigl\Vert \bigl(I-A^{*}B\bigr)^{-1} \bigr\Vert \leq{}& \Vert K \Vert \quad\bigl(\text{by (4)} \bigr) \\ \leq{}& \bigl\Vert \bigl(\bigl(I-A^{*}A\bigr)^{-1}\sharp \bigl(I-B^{*}B\bigr)^{-1}\bigr)+\bigl(\bigl(I-B^{*}B \bigr)^{-1} \sharp \bigl(I-A^{*}A\bigr)^{-1} \bigr) \bigr\Vert \\ & \bigl(\text{by (1)}\bigr) \\ ={}&2 \bigl\Vert \bigl(I-B^{*}B\bigr)^{-1}\sharp \bigl(I-A^{*}A\bigr)^{-1} \bigr\Vert . \end{aligned}$$

 □

Remark 2.9

Obviously, our result is sharper than (2).

Finally, we present an extension of Theorem 1.3.

Theorem 2.10

Let\(A, B, X, Y\in \mathbb{M}_{n}\). IfM=(AXYB)is SPT, then

$$\begin{aligned} s_{j} \biggl(\varPhi \biggl(\frac{X+Y}{2} \biggr) \biggr) \leq s_{j} \bigl(\varPhi (A\sharp B ) \bigr), \end{aligned}$$
(10)

where\(\varPhi: C\rightarrow C+\operatorname{Tr}(C)I\).

Proof

Since M is SPT, then

$$\begin{aligned} \operatorname {Re}M = \begin{pmatrix} \operatorname {Re}A & (X+Y)/2 \\ (X+Y)^{*}/2 &\operatorname {Re}B \end{pmatrix} \end{aligned}$$

and

$$\begin{aligned} \operatorname {Re}\bigl(M^{\tau }\bigr) = \begin{pmatrix} \operatorname {Re}A &(X+Y)^{*}/2 \\ (X+Y)/2 &\operatorname {Re}B \end{pmatrix}=(\operatorname {Re}M)^{\tau } \end{aligned}$$

are both positive semidefinite matrices. Thus, ReM is PPT.

By Theorem 1.3, we have

$$\begin{aligned} s_{j} \biggl(\varPhi \biggl(\frac{X+Y}{2} \biggr) \biggr) \leq s_{j} \bigl(\varPhi \bigl((\operatorname {Re}A)\sharp (\operatorname {Re}B) \bigr) \bigr). \end{aligned}$$

Compute

$$\begin{aligned} s_{j} \biggl(\varPhi \biggl(\frac{X+Y}{2} \biggr) \biggr)&\leq s_{j} \bigl(\varPhi (\operatorname {Re}A\sharp \operatorname {Re}B ) \bigr) \\ &\leq s_{j} \bigl(\varPhi \bigl(\operatorname {Re}(A\sharp B ) \bigr) \bigr) \quad\bigl(\text{by (7)}\bigr) \\ &=s_{j} \bigl(\operatorname {Re}\bigl(\varPhi (A\sharp B)\bigr) \bigr) \\ &\leq s_{j}\bigl(\varPhi (A\sharp B)\bigr) \quad \bigl(\text{by (3)}\bigr). \end{aligned}$$

 □

Remark 2.11

If M is PPT, then (10) becomes Lin’s result in Theorem 1.3.

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The work is supported by Hainan Provincial Natural Science Foundation for High-level Talents grant no. 2019RC171, the Ministry of Education of Hainan grant no. Hnky2019ZD-13, China Scholarship Council grant no. 201908460006, the Ministry of Education of Hainan grant no. Hnky2019ZD-13, the Provincial Key Laboratory, Hainan Normal University grant no. JSKX201904 and National Natural Science Foundation of China grant 11671105.

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Correspondence to Junjian Yang.

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Yang, J., Lu, L. & Chen, Z. Some inequalities related to \(2\times 2\) block sector partial transpose matrices. J Inequal Appl 2020, 90 (2020). https://doi.org/10.1186/s13660-020-02358-0

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MSC

  • 15A45
  • 15A60

Keywords

  • Sector partial transpose matrices
  • Unitarily invariant norms
  • Linear maps