Jean Christophe Bourin, Eun-Young Lee
We obtain several norm and eigenvalue inequalities for positive matrices partitioned into four blocks. The results involve the numerical range of the off-diagonal block X, especially the distance from 0 to W(X).
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Toeplitz matrices are ubiquitous and play important roles across many areas of mathematics. In this paper, we present some algebraic results concerning block Toeplitz matrices with block entries belonging to a commutative algebra $Å$. The characterization of normal block Toeplitz matrices with entries from $Å$ is also obtained.
General approach to the multiplication or adjoint operation of $2\times 2$ block operator matrices with unbounded entries are founded. Furthermore, criteria for self-adjointness of block operator matrices based on their entry operators are established.
The maximal commutative subalgebras containing only Toeplitz matrices have been identified as generalized circulants. A similar simple description cannot be obtained for block Toeplitz matrices. We introduce and investigate certain families of maximal commutative algebras of block Toeplitz matrices.
Several new verifiable conditions are established for block matrices with scalar diagonal blocks to have the numerical range equal the convex hull of at most k ellipses where k by k is the size of the smaller diagonal block. For k = 2, these conditions are also necessary, provided that the ellipses are co-centered.
Positive semidefinite matrices partitioned into a small number of Hermitian blocks have a remarkable property. Such a matrix may be written in a simple way from the sum of its diagonal blocks
Block Toeplitz Matrices: Multiplicative Properties
本文研究了块Toeplitz矩阵的乘法性质，证明了关于乘积$AB-CD$的基本结果，并获得了对角矩阵代数中正规块Toeplitz矩阵在基变换下的刻画。
We prove some eigenvalue inequalities for positive semidefinite matrices partitioned into four blocks. The inradius of the numerical range of the off-diagonal block contributes to these estimates. Some related norm inequalities are given and a conjecture is proposed.
The maximal algebras of scalar Toeplitz matrices are known to be formed by generalized circulants. The identification of algebras consisting of block Toeplitz matrices is a harder problem, that has received little attention up to now. We consider the case when the block entries of the matrices belong to a commutative algebra $ \mathcal{A} $. After obtaining some general results, we classify all the maximal algebras for certain particular cases of $ \mathcal{A}$.
Maximal Algebras of Block Toeplitz Matrices with Entries in the Schur Algebra
在Schur代数中研究块Toeplitz矩阵的最大代数分类，揭示其完整结构与性质。
Krein Space Numerical Range of Block Matrices -- a Unified Approach to the Hyperbolic Case
本文针对对角块为单位矩阵标量倍数的2×2块矩阵，研究其Krein空间数值域在边界生成曲线为双曲线时的情形，提出统一方法以推导数值域双曲形状的已有及新结果。
Let H be a positive semidefinite matrix partitioned into Hermitian blocks. Then, up to a direct sum operation, H is the average of matrices isometrically congruent to its partial trace. A few corollaries are given, related to important inequalities in quantum information theory such as the Nielsen-Kempe separability criterion.
In this short note we deal with a constructive scheme to decompose a continuous family of matrices $A(ρ)$ asymptotically as $ρ\to0$ into blocks corresponding to groups of eigenvalues of the limit matrix A(0). We also discuss the extension of the scheme to matrix families depending upon additional parameters and operators on Hilbert spaces.
The numerical radius and positivity of block matrices
本文以块矩阵的正定性为核心，通过刻画数值半径以一为界的算子，系统阐述块矩阵应用的主要思想与技术，可作为该主题的导论。
Diagonal and off-diagonal blocks of positive definite partitioned matrices
该研究建立了正定分块矩阵块间的精确不等式，并提出关于三个收缩算子的三角型不等式猜想，认为不等式中常数c=3/4是精确的。
Let $T=\begin{bmatrix} X &Y\\ 0 & Z\end{bmatrix}$ be an $n$-square matrix, where $X, Z$ are $r$-square and $(n-r)$-square, respectively. Among other determinantal inequalities, it is proved $\det(I_n+T^*T)\ge \det(I_r+X^*X)\cdot \det(I_{n-r}+Z^*Z)$ with equality holds if and only if $Y=0$.
We bring in some new notions associated with $2\times 2$ block positive semidefinite matrices. These notions concern the inequalities between the singular values of the off diagonal blocks and the eigenvalues of the arithmetic mean or geometric mean of the diagonal blocks. We investigate some relations between them. Many examples are included to illustrate these relations.
Several subadditivity results and conjectures are given for matrices (or operators), block-matrices, concave functions and norms.
It is well known that an operator-valued function $Θ$ from the Schur class ${\bf S}(\mathfrak M,\mathfrak N)$, where $\mathfrak M$ and $\mathfrak N$ are separable Hilbert spaces, can be realized as the transfer function of a simple conservative discrete time-invariant linear system. The known realizations involve the function $Θ$ itself, the Hardy spaces or the reproducing kernel Hilbert spaces. On the other hand, as in the classical scalar case, the Schur class operator-valued function is uniquely determined by its so called "Schur parameters". In this paper we construct simple conservative r…
Let $H$ be a positive semi-definite matrix partitioned in $β\times β$ Hermitian blocks, $H=[A_{s,t}]$, $1\le s,t,\le β$. Then, for all symmetric norms, {equation*} \| H \| \le \| \sum_{s=1}^β A_{s,s} \|. {equation*} The proof uses a nice decomposition for positive matrices and unitary congruences with the generators of a Clifford algebra. A few corollaries are given, in particular the partial trace operation increases norms of separable states on a real Hilbert space, leading to a conjecture for usual complex Hilbert spaces.
It is known that every complex square matrix with nonnegative determinant is the product of positive semi-definite matrices. There are characterizations of matrices that require two or five positive semi-definite matrices in the product. However, the characterizations of matrices that require three or four positive semi-definite matrices in the product are lacking. In this paper, we give a complete characterization of these two types of matrices. With these results, we give an algorithm to determine whether a square matrix can be expressed as the product of $k$ positive semi-definite matrices …
We study the class of Hankel matrices for which the $k\times k$-block-matrices are positive semi-definite. We prove that a $k\times k$-block-matrix has non zero determinant if and only if all $k\times k$-block matrices have non zero determinant. We use this result to extend the notion of propagation phenomena to $k$-hyponormal weighted shifts. Finally we give a study on invariance of $k$-hyponormal weighted shifts under one rank perturbation.
We conjecture the following so-called norm compression inequality for $2\times N$ partitioned block matrices and the Schatten $p$-norms: for $p\ge 2$, $$ ||({array}{cccc} A_1 & A_2 & ... & A_N B_1 & B_2 & >... & B_N {array})||_p \le ||({array}{cccc} ||A_1||_p & ||A_2||_p & ... & ||A_N||_p \ ||B_1||_p & ||B_2||_p & ... & ||B_N||_p {array})||_p $$ while for $1\le p\le 2$ the ordering of the inequality is reversed. This inequality includes Hanner's inequality for matrices as a special case. We prove several special cases of this inequality and give examples for $3\times 3$ and larger partitioning…
We show how positive unital linear maps can be used to obtain some bounds for the eigenvalues of nonnegative matrices.
We consider a class of block operator matrices arising in the study of scattering passive systems, especially in the context of boundary control problems. We prove that these block operator matrices are indeed a subclass of block operator matrices considered in [Trostorff: A characterization of boundary conditions yielding maximal monotone operators. J. Funct. Anal., 267(8): 2787--2822, 2014], which can be characterized in terms of an associated boundary relation.
This short note, in part of expository nature, points out several new or recent consequences of a quite nice decomposition for positive semi-definite matrices.
The CMV matrices and their sub-matrices are applied to the description of all solutions to the Schur interpolation problem for contractive analytic operator-valued functions in the unit disk (the Schur class functions).
We review some of the significant generalizations and applications of the celebrated Douglas theorem on the equivalence of factorization, range inclusion, and majorization of operators. We then apply it to find a characterization of the positivity of $2\times 2$ block matrices of operators in Hilbert spaces and finally describe the nature of such block matrices and provide several ways for showing their positivity.
The main purpose of this paper is to englobe some new and known types of Hermitian block-matrices $M=\begin{pmatrix} A \& X\\ {X^*} \& B\end{pmatrix}$ satisfying or not the inequality $\|M\|\le \|A+B\|$ for all symmetric norms
An Oppenheim type inequality for positive definite block matrices
本文针对正定分块矩阵提出一个新的 Oppenheim 型行列式不等式，旨在去除 Lin 结果中两个矩阵必须可交换的限制，给出更自然的推广。
In this paper, we study a particular class of block matrices placing an emphasis on their spectral properties. Some related applications are then presented.
For positive semi-definite block-matrix $M,$ we say that $M$ is P.S.D. and we write $M=\begin{pmatrix} A \& X\\ {X^*} \& B\end{pmatrix} \in {\mathbb{M}}\_{n+m}^+$, with $A\in {\mathbb{M}}\_n^+$, $B \in {\mathbb{M}}\_m^+.$ The focus is on studying the consequences of a decomposition lemma due to C.~Bourrin and the main result is extending the class of P.S.D. matrices $M$ written by blocks of same size that satisfies the inequality: $\|M\|\le \|A+B\|$ for all symmetric norms.
On the inversion of the block double-structured and of the triple-structured Toeplitz matrices and on the corresponding reflection coefficients
本文在已有二维TBT矩阵求逆刻画基础上，将方法推广到分块TBT矩阵与三维Toeplitz矩阵，研究其逆矩阵恢复所需最小信息及相应反射系数。
We study basic spectral properties of J-self-adjoint $2\times 2$ block operator matrices. Using the linear resolvent growth condition, we obtain simple necessary conditions for the regularity of the critical point $\infty$. In particular, we present simple examples of operators having the singular critical point $\infty$. Also, we apply our results to the linearized operator arising in the study of soliton type solutions to the nonlinear relativistic Ginzburg-Landau equation.
The results on the inversion of convolution operators and Toeplitz matrices in the 1-D (one dimensional) case are classical and have numerous applications. We consider a 2-D case of Toeplitz-block Toeplitz matrices, describe a minimal information, which is necessary to recover the inverse matrices, and give a complete characterisation of the inverse matrices.
We study the spectral properties of bounded and unbounded Jacobi matrices whose entries are bounded operators on a complex Hilbert space. In particular, we formulate conditions assuring that the spectrum of the studied operators is continuous. Uniform asymptotics of generalised eigenvectors and conditions implying complete indeterminacy are also provided.
Mesoscopic Spectral CLT for Block Correlated Random Matrices
本文研究了块相关随机矩阵的线性特征值统计量涨落，利用矩阵值 Dyson 方程的两点版本证明其在所有介观尺度上呈现高斯性，且方差具有与高斯正交或酉系综一致的普适性。
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