Related papers: Two trace inequalities for operator functions

In this note we study the problem of evaluating the trace of $f(T)-f(R)$, where $T$ and $R$ are contractions on Hilbert space with trace class difference, i.e., $T-R\in\boldsymbol{S}_1$ and $f$ is a function analytic in the unit disk ${\Bbb…

For positive definite matrices $A$ and $B$, the Araki-Lieb-Thirring inequality amounts to an eigenvalue log-submajorisation relation for fractional powers $$\lambda(A^t B^t) \prec_{w(\log)} \lambda^t(AB), \quad 0<t\le 1,$$ while for…

Let $A,B\in B(H)$. We present among others a simple proof of the widely known result stating that if $0\leq A\leq B$, then $\sqrt A\leq \sqrt B$. The same idea is used to prove that if $0\leq A\leq B$ and $A$ is invertible, then $B$ too is…

In \cite{AP2} we obtained general estimates of the operator moduli of continuity of functions on the real line. In this paper we improve the estimates obtained in \cite{AP2} for certain special classes of functions. In particular, we…

We improve and generalize some operator inequalities for positive linear maps. It is shown, among other inequalities, that if $0<m\le B\le m'<M'\le A\le M$ or $0<m\le A\le m'<M'\le B\le M$, then for each $2\le p<\infty $ and $\nu \in \left[…

We prove an operator inequality that extends strong subadditivity of entropy: after taking a trace, the operator inequality becomes the strong subadditivity of entropy.

Given a function $f: (a,b) \rightarrow \mathbb{R},$ L\"owner's theorem states $f$ is monotone when extended to self-adjoint matrices via the functional calculus, if and only if $f$ extends to a self-map of the complex upper half plane. In…

The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that Rockafellar's constraint qualification holds. In this paper, we prove the maximal…

We prove that if $f$ is a Lipschitz function on $\R$, $A$ and $B$ are self-adjoint operators such that ${\rm rank} (A-B)=1$, then $f(A)-f(B)$ belongs to the weak space $\boldsymbol{S}_{1,\be}$, i.e., $s_j(A-B)\le{\rm const} (1+j)^{-1}$. We…

In this paper, we present some operator and eigenvalue inequalities involving operator monotone, doubly concave and doubly convex functions. These inequalities provide some variants of operator Acz\'{e}l inequality and its reverse via…

We prove some refinements of a reverse AM-GM operator inequality due to M. Lin [Studia Math. 2013;215:187-194]. In particular, we show the operator inequality \begin{eqnarray*} \Phi^p\left(A\nabla_\nu B+2rMm(A^{-1}\nabla B^{-1}-A^{-1}\sharp…

We describe recent work of Kim in arXiv:1210.5190 to show that operator convex functions associated with quasi-entropies can be used to prove a large class of new matrix inequalities in the tri-partite and bi-partite setting by taking a…

For a given $\delta$, $0<\delta<1$, a Blaschke sequence $\sigma=\{\lambda_j\}$ is constructed such that every function $f$, $f\in H^\infty$, having $\delta<\delta_f=\inf_{\lambda\in\sigma}|f(\lambda)|\le\|f\|_\infty\le1$ is invertible in…

We present several sharp upper bounds and some extension for product operators. Among other inequalities, it is shown that if , , are non-negative continuous functions on such that , , then for all non-negative operator monotone decreasing…

Let $\Phi$ be a unital positive linear map and let $A$ be a positive invertible operator. We prove that there exist partial isometries $U$ and $V$ such that \[ |\Phi(f(A))\Phi(A)\Phi(g(A))|\leq U^*\Phi(f(A)Ag(A))U \] and…

We propose a notion of operator monotonicity for functions of several variables, which extends the well known notion of operator monotonicity for functions of only one variable. The notion is chosen such that a fundamental relationship…

This paper formulates Young-type inequalities for singular values (or $s$-numbers) and traces in the context of von Neumann algebras. In particular, it shown that if $\t(\cdot)$ is a faithful semifinite normal trace on a semifinite von…

Let $A$ be a positive semidefinite $m\times m$ block matrix with each block $n$-square, then the following determinantal inequality for partial traces holds \[ (\mathrm{tr} A)^{mn} - \det(\mathrm{tr}_2 A)^n \ge \bigl| \det A -…

Associativity of a two-place function $T: [0,1]^2\rightarrow [0,1]$ defined by $T(x,y)=f^{(-1)}(F(f(x),f(y)))$ where $F:[0,\infty]^2\rightarrow[0,\infty]$ is an associative function, $f: [0,1]\rightarrow [0,\infty]$ is a monotone function…

This article improves the triangle inequality for complex numbers, using the Hermite-Hadamard inequality for convex functions. Then, applications of the obtained refinement are presented to include some operator inequalities. The operator…