Related papers: Problems and Conjectures in Matrix and Operator In…
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.
In recent years, random matrices have come to play a major role in computational mathematics, but most of the classical areas of random matrix theory remain the province of experts. Over the last decade, with the advent of matrix…
In this note devoted to some aspects of the inverse problem of representation theory the attention is concentrated on the interrelations between various algebraic structures (algebras with operators) unraveled by different solutions of the…
In [P. Renaud, "A matrix formulation of Gr\"uss inequality", Linear Algebra Appl. 335 (2001), 95--100] it was proved an operator inequality involving the usual trace functional. In this article, we give a refinement of such result and we…
Operator matrices have played a significant role in studying Hilbert space operators. In this paper, we discuss further properties of operator matrices and present new estimates for the operator norms and numerical radii of such operators.…
We explore the uncertainty relation for unitary operators in a new way and find two uncertainty equalities for unitary operators, which are minimized by any pure states. Additionally, we derive two sets of uncertainty inequalities that…
This is a survey on algorithmic questions about combinatorial and geometric properties of convex polytopes. We give a list of 35 problems; for each the current state of knowledege on its theoretical complexity status is reported. The…
Three open problems in analysis are formulated: 1. Injectivity of Radon transform 2. A uniqueness problem for an elliptic system of PDE 3. A problem in operator theory
We revisit the matrix problems sparse null space and matrix sparsification, and show that they are equivalent. We then proceed to seek algorithms for these problems: We prove the hardness of approximation of these problems, and also give a…
In this note we briefly survey and propose some open problems related to isoparametric theory.
Several inequalities for eigenvalues involving convex combinations and compressions are given. These inequalities are matrix version of the basic convexity inequality f((a+b)/2) < (f(a)+f(b))/2.
We present an inequality for tensor product of positive operators on Hilbert spaces by considering the tensor product of operators as words on certain alphabets (i.e., a set of letters). As applications of the operator inequality and by a…
We will consider about some inequalities on operator means for more than three operators, for instance, ALM and BMP geometric means will be considered. Moreover, log-Euclidean and logarithmic means for several operators will be treated.
Operator learning offers a robust framework for approximating mappings between infinite-dimensional function spaces. It has also become a powerful tool for solving inverse problems in the computational sciences. This chapter surveys…
Random matrices now play a role in many parts of computational mathematics. To advance these applications, it is desirable to have tools that are flexible, easy to use, and powerful. Over the last 25 years, researchers have developed a…
In this survey, we shall present characterizations of some distinguished classes of Hilbertian bounded linear operators (namely, normal operators, selfadjoint operators, and unitary operators) in terms of operator inequalities related to…
Over 300 sequences and many unsolved problems and conjectures related to them are presented herein. These notions, definitions, unsolved problems, questions, theorems corollaries, formulae, conjectures, examples, mathematical criteria, etc.…
New sharp multiplicative reverses of the operator means inequalities are presented, with a simple discussion of squaring an operator inequality. As a direct consequence, we extend the operator P\'olya-Szeg\"o inequality to arbitrary…
Some rearrangement inequalities for symmetric norms on matrices are given as well as related results for operator convex functions.
Mercer inequality for convex functions is a variant of Jensen's inequality, with an operator version that is still valid without operator convexity. This paper is two folded. First, we present a Mercer-type inequality for operators without…