Two trace inequalities for operator functions

Convexity and matrix means

A capital letter means a bounded liner operator on a complex Hilbert space H , and B(H) denotes the algebra of all bounded linear operators on H equipped with operator norm ‖ · ‖ . An operator T is said to be positive (denoted by T 0) if (Tx,x) 0 for all x∈H and an operator T is said to be strictly positive (denoted by T > 0) if T is positive and invertible. Chaotic order is defined by logA logB for strictly positive operators A and B . A continuous real-valued function f defined on an interval J is called operator monotone if A B implies that f (A) f (B) for all self-adjoint operators A , B with spectra in J . The well known celebrated Lower-Heinz inequality asserts that if A B 0, then Aα Bα for any α ∈ [0,1] , which means that t → tα is operator monotone. Another well known example of operator monotone function is t → log t on (0,∞) . Recently, Furuta [5] showed that if A > B , then f (Aα ) > f (Bα) for all α ∈ (0,1] and if logA > logB , then there exists β ∈ (0,1] such that f (Aα) > f (Bα) for all α ∈ (0,β ] , where f (t) is a non-constant operator monotone function on [0,∞) . Comprehensive survey on related order preserving operator inequalities were given in [2,4]. In this paper, we will obtain an improvement which leads to the result of Moslehian and Najafi [6].

Extreme rays for cones of Hermitian matrix functions that are non-negative on the positive semi-definite matrices

Numerous results are presented that characterize when a complex Hermitian matrix is Birkhoff-James orthogonal, in the trace norm, to a (Hermitian) positive semidefinite matrix or set of positive semidefinite matrices. For example, a simple-to-test criterion that determines which Hermitian matrices are Birkhoff-James orthogonal, in the trace norm, to the set of all positive semidefinite diagonal matrices is developed. Applications in the theory of quantum resources are explored. For example, the quantum states that have modified trace distance of coherence equal to $1$ (the maximal possible value) are characterized, and a connection between the modified trace distance of $2$-entanglement and the NPPT bound entanglement problem is established.

Sparse matrix decompositions and graph characterizations

The explicit form of Harnack’s inequality for non-negative harmonic functions in the open ball plays an important role in harmonic analysis and elliptic partial differential equations. In this paper, we establish the explicit form of Harnack’s inequality for Poisson’s Dirichlet problem in the open ball with non-negative boundary data. We involve the Green’s function for Laplacian operator to find it out. Harnack’s inequality for positive harmonic functions can be followed from Harnack’s inequality for Poisson’s Dirichlet problem when the source function is set to be zero.

On vector configurations that can be realized in the cone of positive matrices

On submodular prophet inequalities and correlation gap

Solving symmetric matrix word equations via symmetric space machinery

Assume that two sets of k vectors in Rn are given, namely {x1,...,xk} and {y1,...,yk}, and a class of matrices, e.g., positive definite matrices, positive matrices, strictly totally positive matrices, or P-matrices. The question considered in this paper is that of determining necessary and sufficient conditions on these sets of vectors such that there exists an n x n matrix A in the given class satisfying Axj=yj (j= 1,...,k).

We study matrix inequalities involving partial traces for positive semidefinite block matrices. First of all, we present a new method to prove a celebrated result of Choi [Linear Algebra Appl. 516 (2017)]. The method also allows us to prove a generalization of another result of Choi [Linear Multilinear Algebra 66 (2018)]. Furthermore, we shall give an improvement on a recent result of Li, Liu and Huang [Operators and Matrices 15 (2021)]. In addition, we include with some majorization inequalities involving partial traces for two by two block matrices, and also provide inequalities related to the unitarily invariant norms as well as the singular values, which can be viewed as slight extensions of two results of Lin [Linear Algebra Appl. 459 (2014)] and [Electronic J Linear Algebra 31 (2016)].

Trace inequalities for multiple products of powers of two matrices are discussed via the methodof logmajorization. For instance, the trace inequality |Tr(Ap1Bq1Ap2Bq2 · · ·ApK BqK )| Tr(AB) is obtained for positive semidefinite matrices A,B and pi, qi 0 with p1 + · · ·+pK = q1 + · · · + qK = 1 under some additional condition. Mathematics subject classification (1991): 15A45, 15A42, 15A60, 47A30. Keywordsandphrases: Positive semidefinitematrices, trace inequalities, Golden-Thompson inequality, log majorization.

A Note on Solution of the Nearest Correlation Matrix Problem by von Neumann Matrix Divergence

Necessary conditions are developed for a system of the form Aλ=B with A≥0 and B≥ to have a least squares solution X which is nonnegative. Also it is shown that if a nonnegative matrix A has a nonnegative W-weighted { 1,3 -inverse for some nonnegative positive definite symmetric matrix W. then A has a nonnegative (1, 3) -inverse. As, a consequence of this, a short proof is obtained of a recent theorem of Jain and Egawa concerning nonnegative best approximate solutions (SIAM J. ALG. DISC. METH., 3 (1982), 197-213).

Geometric mean block matrices