Some inequalities related to 2×2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2\times 2$\end{document} block sector partial transpose matrices

In this article, two inequalities related to 2×2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2\times 2$\end{document} 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 M n the set of n × n complex matrices. M n (M k ) is the set of n × n block matrices with each block in M k . The n × n identity matrix is denoted by I n . We use · for an arbitrary unitarily invariant norm. A positive semidefinite matrix A will be expressed as A ≥ 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), . . . , 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) ≥ s 2 (A) ≥ · · · ≥ s n (A). When A is Hermitian, we enumerate eigenvalues of A in nonincreasing order σ 1 (A) ≥ σ 2 (A) ≥ · · · ≥ σ n (A). Recall that C ∈ M m×n is (strictly) contractive if (I n > C * C) I n ≥ C * C. The geometric mean of two positive definite matrices A, B ∈ M n , denoted by A B, is the positive definite solution of the Riccati equation More details on the matrix geometric mean can be found in [2,Chap. 4].
The numerical range of A ∈ M n is defined by For basic properties of numerical range, see [5]. Also, we define a sector on the complex Actually, the class of matrices T with W (T) ⊆ S α 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].
is said to be positive partial transpose (i.e. PPT) if H is positive semidefinite and its partial transpose H τ = (H ji ) n j,i=1 is also positive semidefinite. Inspired by PPT, Kuai [6] defined a new conception called sectorial partial transpose (i.e.
Thus, it is natural to extend the results for PPT matrices to SPT matrices.
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
where A, B ∈ M m×n are strictly contractive, then 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.
is PPT, then for the linear map Φ : C → C + Tr(C)I, 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.
The next lemma is due to Zhang [14, Lemma 3.1].

Lemma 2.4 Let
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 ∈ M n be matrices with positive semidefinite real part. Then
Now we are ready to present our results. The first theorem is an extension of Theorem 1. Next will give a stronger inequality than Theorem 1.2.

Theorem 2.8 Let the Hua matrix be given by
where A, B ∈ M m×n are strictly contractive. Then for any unitarily invariant norm.
Proof Since H is PPT, then are both positive semidefinite matrices. Hence, Clearly, by unitary similarity transformation, is also positive semidefinite. Therefore, Thus, by (5) and monotonicity by (8) and (9) = 0.
In a similar way, we can prove 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 ∈ M n . If M = A X Y * B is SPT, then where Φ : C → C + Tr(C)I.

Proof
Since M is SPT, then Compute