Yassine Bedrani, Fuad Kittaneh, Mohammed Sababheh
In this paper, we discuss new inequalities for accretive matrices through non standard domains. In particular, we present several relations for $A^r$ and $A\sharp_rB$, when $A,B$ are accretive and $r\in (-1,0)\cup (1,2).$ This complements the well established discussion of such quantities for accretive matrices when $r\in [0,1],$ and provides accretive versions of known results for positive matrices.
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The concept of Loewner (partial) order for general complex matrices is introduced. After giving the definition of arithmetic, geometric, and harmonic mean for accretive-dissipative matrices, we study their basic properties. An AM-GM-HM inequality is established for two accretive-dissipative matrices in the sense of this extended Loewner order. We also compare the harmonic mean and parallel sum of two accretive-dissipative matrices, revealing an interesting relation between them. A number of examples are included.
The main goal of this paper is to discuss the recent advancements of operator means for accretive matrices in a more general setting. In particular, we present the general form governing the well established definition of geometric mean, then we define arbitrary operator means and functional calculus for accretive matrices. Applications of this new discussion involve generalizations of known inequalities from the setting of positive matrices to that of accretive matrices. This includes the arithmetic-harmonic mean comparisons, monotony of operator means, Ando's inequality, Choi's inequality, A…
The numerical radius of a matrix is a scalar quantity that has many applications in the study of matrix analysis. Due to the difficulty in computing the numerical radius, inequalities bounding it have received a considerable attention in the literature. In this article, we present many new bounds for the numerical radius of accretive matrices. The importance of this study is the presence of a new approach that treats a specific class of matrices, namely the accretive ones. The new bounds provide a new set of inequalities, some of which can be considered as refinements of other existing ones, w…
Let $A={bmatrix} A_{11} &A_{12} A_{21} & A_{22} {bmatrix}$ be an $n\times n$ accretive-dissipative matrix, $k$ and l be the orders of $A_{11}$ and $A_{22}$, respectively, and let $m=\min\{k,l\}$. Then $$|\det A|\le a|\det A_{11}|\cdot|\det A_{22}|,$$ where $a=\{{array}{l l} 2^{3m/2}, & \text{if} m\le n/3; 2^{n/2}, & \text{if} n/3<m\le n/2. {array}.$ This improves a result of Ikramov.
Weighted numerical radii of accretive matrices
本文研究 accretive 矩阵的加权数值半径 $\u03c9_{\u03bd}(A)$，其推广了经典数值半径与算子范数，且当 $\u03bd=1/2$ 或矩阵正定时，所得不等式可退化为经典形式。
To better understand the algebra $\mathcal{M}_n$ of all $n\times n$ complex matrices, we explore the class of accretive matrices. This class has received renowned attention in recent years due to its role in complementing those results known for positive definite matrices. More precisely, we have several results that allow a better understanding of accretive matrices. Among many results, we present order-preserving results, Choi-Davis-type inequalities, mean-convex inequalities, sub-multiplicative results for the real part, and new bounds of the absolute value of accretive matrices. These resu…
Some Hadamard product inequalities for accretive matrices
本文研究增生矩阵的 Hadamard 乘积不等式，结合矩阵均值、正单位线性映射与矩阵凹函数，证明了正定矩阵在特定幂指数范围内的新不等式。
Accretive Partial Transpose Matrices and Their Connections to Matrix Means
本文研究增生部分转置（APT）矩阵的性质，将其推广自PPT矩阵，并证明在扇形矩阵和算子单调函数条件下，由加权算术均值构造的块矩阵保持APT性质。
Schrödinger operators with accretive potentials in weighted spaces
本文研究加权空间中带增性势的薛定谔算子，给出预解集非空、定义域分离及预解紧性与Schatten类条件，并推广特征系统完备性至加权空间，应用于强增性阻尼波动方程。
The numerical radius of fractional powers of matrices
本文利用分数次幂积分表示与扇形矩阵几何性质，证明增生-耗散矩阵满足 ω(A^t)≥ω^t(A)，并推广到增生矩阵在 0<t<1/2 时的情形。
Functions and Means of Accretive Operators
研究希尔伯特空间上增生算子的函数与均值，通过里斯 - 邓福德积分定义\(f(A)\)并推广分数幂，将几何均值等推广到增生算子，给出了\(A\# B\)及\(A\# A^*\)等相关的等式与不等式关系。
Johnson's determinantal identity for contiguous minors of Toeplitz matrices, with an accretive extension
本文证明了托普利茨矩阵连续小矩阵的行列式恒等式，并扩展了贝亚特-蒂摩里算术-几何平均恒等式，通过多项式恒等式和秩为一的条件得出不等式。
In this article, we employ a standard convex argument to obtain new and refined inequalities related to the matrix mean of two accretive matrices, the numerical radius and the Tsallis relative operator entropy.
The Resolvent Mean and The Parametrized $\mathcal{A} \sharp \mathcal{B}$
本文将正定矩阵的预解平均与加权 $\mathcal{A}\sharp \mathcal{H}$-平均概念推广至正定矩阵，探讨了其性质，并建立了预解平均与加权几何平均之间的新不等式关系。
A note on $l^p$ norms of weighted mean matrices
本文给出了加权平均矩阵的$l^p$范数的一些结果。这些结果可以看作是类似于Bennett关于加权Carleman不等式的一个结果。
A note on $l^2$ norms of weighted mean matrices
本文证明了Cartlidge关于加权平均矩阵的$l^{p}$算子范数的一个结果，其中$p=2$的范数是某些矩阵的特征值.
Operator mean inequalities for sector matrices
本研究针对扇区矩阵的算子均值问题，推广并精细化了已有不等式结果，推导了含正单式线性映射、伴随均值与Kantorovich常数的新不等式，丰富了算子均值与扇区矩阵的理论研究。
On l^p norms of weighted mean matrices
利用Kaluza-Szegö和Redheffer的方法研究了加权平均矩阵的$l^{p}$算子范数.作为应用，证明了Bennett的一个猜想.
On $l^2$ norms of some weighted mean matrices
本文给出了Bennett关于$p=2$情形下某些加权平均矩阵的$l^{p}$算子范数的一个结果的另一个证明，并给出了一些相关结果.
On weighted mean matrices whose $l^p$ norms are determined on decreasing sequences
本文给出了加权平均矩阵的一个条件，使得当满足该条件时，其$l^p$范数在降序列上确定。应用我们的结果证明了Bennett的一个猜想，并讨论了相关的结果.
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 study the $l^p$ norms of a class of weighted mean matrices whose diagonal terms are given by $n^α/\sum^{n}_{i=1}i^α$ with $α> -1$. The $l^p$ norms of such matrices are known for $p \geq 2, (α+1)p >1$ and $1<p \leq 4/3, 1/p \leq α\leq 1$. In this paper, we determine the $l^p$ norms of such matrices for $p \geq 1.35, 0\leq α\leq 1$.
The concepts of weighted numerical radius has been defined in recent times. In this article, we obtain several upper bound for weighted numerical radius of operators and $2 \times 2$ operator matrices which generalize and improves some well known famous inequality for classical numerical radius. We also obtain an upper bound for the weighted numerical radius of the Aluthge transformation, $\tilde{T}$ of an operator $T \in \mathcal{B}(\mathcal{H}),$ where $\tilde{T} = |T|^{1/2} U |T|^{1/2}$ and $T = U |T|$ be the canonical polar decomposition of $T.$
Rhaly operators on weighted Hardy spaces and factorable matrices
研究加权Hardy空间上Rhaly算子的性质，通过可分解矩阵刻画其Schatten类。
Let $μ_p(A,B,t)=(tA^p+(1-t)B^p)^{1/p}$ denote the weighted power mean between positive operators $A$ and $B$. We show that the function $t\to \|A-μ_p(A,B,t)\|_2$ is monotonically decreasing whenever $1/2 \leq p \leq 1$. Hence showing that the weighted power means satisfy Audenaert's "in-betweenness" property for positive operators for power satisfying $1/2 \leq p \leq 1$. We also show that when $p>2$ there exist operators for which the weighted power mean does not satisfy this "in-betweenness" property with respect to the Euclidean metric.
In this article, we are going to introduce the weighted numerical range which is a further generalization both the c-numerical range and the rank k numerical range. If the boundaries of weighted numerical ranges of two matrices (possibly of different sizes) overlap at sufficiently many points, then the two matrices share common generalized eigenvalues.
The main aim of this paper is to compare two recent approaches for investigating the interspace between the union of Gevrey spaces $\mathcal G_t (U)$ and the space of smooth functions $C^{\infty}(U)$. The first approach in the style of Komatsu is based on the properties of two parameter sequences $M_p=p^{τp^σ}$, $τ>0$, $σ>1$. The other one uses weight matrices defined by certain weight functions. We prove the equivalence of the corresponding spaces in the Beurling case by taking projective limits with respect to matrix parameters, while in the Roumieu case we need to consider a larger space th…
On inclusion relations of weighted $L^p$-type spaces defined in terms of weight function matrices
本文引入由权重函数矩阵定义的加权$L^p$型空间，研究其包含关系，并探讨其与常规$L^p$空间的重合性、加权空间的非平凡性及平移不变性，最终应用于特定权重函数矩阵。
In this paper the geometric mean of partial positive definite matrices with missing entries is considered. The weighted geometric mean of two sets of positive matrices is defined, and we show whether such a geometric mean holds certain properties which the weighted geometric mean of two positive definite matrices satisfies. Additionally, counterexamples demonstrate that certain properties do not hold. A Loewner order on partial Hermitian matrices is also defined. The known results for the maximum determinant positive completion are developed with an integral representation, and the results are…
We consider in this paper two different types of the weighted geometric means of positive definite operators. We show the component-wise bijection of these geometric means and give a geometric property of the spectral geometric mean as a metric midpoint. Moreover, several interesting inequalities related with the geometric means of positive definite operators will be shown. We also see the meaning of weighted geometric means in the gyrogroup structure with finite dimension and find the formulas of weighted geometric means of 2-by-2 positive definite matrices and density matrices.
We offer a new proof of uniform convexity inequalities for the Finsler manifold of nonpositive curvature taken on the space of positive-semidefinite matrices with the weighted matrix geometric mean defining the geodesic between two points. Using the technique of log majorization, we are able to characterize that the equality cases of said equalities occur if and only if the matrices commute, and hence are the same as in $\ell^p$.
Minimizing numerical radius of weighted cyclic matrices under permutation of the weights
本文研究在固定权重下通过排列权重最小化加权循环矩阵的数值半径，分析不同排列对数值半径波动的影响，探讨最小化策略。
An $n$-by-$n$ ($n\ge 3$) weighted shift matrix $A$ is one of the form $$[{array}{cccc}0 & a_1 & & & 0 & \ddots & & & \ddots & a_{n-1} a_n & & & 0{array}],$$ where the $a_j$'s, called the weights of $A$, are complex numbers. Assume that all $a_j$'s are nonzero and $B$ is an $n$-by-$n$ weighted shift matrix with weights $b_1,..., b_n$. We show that $B$ is unitarily equivalent to $A$ if and only if $b_1... b_n=a_1...a_n$ and, for some fixed $k$, $1\le k \le n$, $|b_j| = |a_{k+j}|$ ($a_{n+j}\equiv a_j$) for all $j$. Next, we show that $A$ is reducible if and only if $A$ has periodic weights, that …
We study $l^p$ operator norms of factorable matrices and related results. We give applications to $l^p$ operator norms of weighted mean matrices and Copson's inequalities. We also apply the method in this paper to study the best constant in an inequality of Hardy, Littlewood and Pólya.
The algebra of Schur operators on l^2 is known not to be inverse-closed. When l^2=l^2(X) where X is a metric space, we can consider elements of the Schur algebra with certain decay at infinity. For instance if X has the doubling property, then Q. Sun has proved that the weighted Schur algebra for a strictly polynomial weight is inverse-closed. Here, we prove a result dealing with left-invertibility. Namely, if such an operator is bounded below in l^p for some p, then it is bounded below for all q, and it admits a left-inverse in the weighted Schur algebra.
Here we prove the convergence of the Ando-Li-Mathias and Bini-Meini-Poloni procedures for matrix means. Actually it is proved here that for a two-variable function which maps pairs of positive definite matrices to a positive definite matrix and is not greater than the square mean of two positive definite matrices, the Ando-Li-Mathias and Bini-Meini-Poloni procedure converges. In order to be able to set up the Bini-Meini-Poloni procedure, a weighted two-variable matrix mean is also needed. Therefore a definition of a two-variable weighted matrix mean corresponding to every symmetric matrix mean…
New perspectives on operator radius bounds in $A-$weighted frameworks
该研究利用Moore-Penrose逆推导半希尔伯特空间算子和的数值半径与范数的界，建立n×n及2×2算子矩阵的A-Davis-Wielandt半径新不等式，得到算子交换子相关A-数值半径不等式的改进上界。
The (un)weighted stability for some matrices is one of essential hypotheses in time-frequency analysis and applied harmonic analysis. In the first part of this paper, we show that for a localized matrix in a Beurling algebra, its weighted stabilities for different exponents and Muckenhoupt weights are equivalent to each other, and reciprocal of its optimal lower stability bound for one exponent and weight is controlled by a polynomial of reciprocal of its optimal lower stability bound for another exponent and weights. Inverse-closed Banach subalgebras of matrices with certain off-diagonal deca…
Invariant embeddings and weighted permutations
该研究证明了酉矩阵共轭下不变的阿贝尔自伴矩阵代数可嵌入同不变性的极大阿贝尔自伴代数，并以此分析特定可交换矩阵的结构、刻画与加权置换酉等价的矩阵。
A New Proof that the Numerical Range is a Complete 2-Spectral Set for Weighted Shift Matrices
本文为 Daeshik Choi 研究的一类加权移位矩阵满足 Crouzeix 猜想提供了新的替代证明，并进一步证明其满足完全有界版本的猜想。
The Karcher Mean on Lie Groups of Upper Triangular Matrices
研究上三角矩阵李群上的加权Karcher均值，利用分差和Parlett递推证明其存在性、唯一性与解析性，并给出递推计算及低维显式公式，揭示分差与矩阵均值的关系。
Let $G$ be a connected graph with $V(G)=\{1,\dotsc,n\}$. Then the resistance distance between any two vertices $i$ and $j$ is given by $r_{ij}:=l_{ii}^† + l_{jj}^†-2 l_{ij}^†$, where $l_{ij}^†$ is the $(i,j)^{\rm th}$ entry of the Moore-Penrose inverse of the Laplacian matrix of $G$. For the resistance matrix $R:=[r_{ij}]$, there is an elegant formula to compute the inverse of $R$. This says that \[R^{-1}=-\frac{1}{2}L + \frac{1}{τ' R τ} ττ', \] where \[τ:=(τ_1,\dotsc,τ_n)'~~\mbox{and}~~ τ_{i}:=2- \sum_{\{j \in V(G):(i,j) \in E(G)\}} r_{ij}~~~i=1,\dotsc,n. \] A far reaching generalization of t…
We introduce a method and an algorithm for computing the weighted Moore-Penrose inverse of multiple-variable polynomial matrix and the related algorithm which is appropriated for sparse polynomial matrices. These methods and algorithms are generalizations of algorithms developed in [M.B. Tasic, P.S. Stanimirovic, M.D. Petkovic, Symbolic computation of weighted Moore-Penrose inverse using partitioning method, Appl. Math. Comput. 189 (2007) 615-640] to multiple-variable rational and polynomial matrices and improvements of these algorithms on sparse matrices. Also, these methods are generalizatio…
We propose a method and algorithm for computing the weighted Moore-Penrose inverse of one-variable rational matrices. Continuing this idea, we develop an algorithm for computing the weighted Moore-Penrose inverse of one-variable polynomial matrix. These methods and algorithms are generalizations of the method for computing the weighted Moore-Penrose inverse for constant matrices, originated in Wang and Chen [G.R. Wang, Y.L. Chen, A recursive algorithm for computing the weighted Moore-Penrose inverse AMN, J. Comput. Math. 4 (1986) 74-85], and the partitioning method for computing the Moore-Penr…
Log-majorizations between quasi-geometric type means for matrices
本文研究了矩阵准几何型均值之间的对数支配关系，并探讨了相关迹函数的联合凹凸性。
Eigenvalues of the laplacian matrices of the cycles with one weighted edge
本文研究带一条加权边的循环图拉普拉斯矩阵的特征值，证明特征值仅依赖于权重实部，给出其在子区间内的定位、牛顿法收敛性及一致误差的渐近公式。
About the generalized LM-inverse and the weighted Moore-Penrose inverse
本文提出了一种递归计算广义LM逆的算法，并证明该算法与Wang和Chen提出的加权Moore-Penrose逆算法等价，通过MATHEMATICA实现并比较了CPU时间。
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