Related papers: Klein's trace inequality and superquadratic trace …

In this paper we show that for a non-negative operator monotone function $f$ on $[0, \infty)$ such that $f(0)= 0$ and for any positive semidefinite matrices $A$ and $B$, $$ Tr((A-B)(f(A)-f(B))) \le Tr(|A-B|f(|A-B|)). $$ When the function…

In this paper, we prove a trace inequality $\text{Tr}[ f(A) A^s B^s ] \leq \text{Tr}[ f(A) (A^{1/2} B A^{1/2} )^s ]$ for any positive and monotone increasing function $f$, $s\in[0,1]$, and positive semi-definite matrices $A$ and $B$. On the…

To each real continuous function f there is an associated trace function on real symmetric matrices Tr f. The classical Klein lemma states that f is convex if and only if Tr f is convex. In this note we present an algebraic strengthening of…

Let f be a function defined on positive numbers. The subject is the trace inequality $Tr f(A) + Tr f(P_2AP_2) \le Tr f(P_{12}AP_{12}) + \Tr f(P_{23}AP_{23})$, where $A$ is a positive operator, $P_1,P_2,P_3$ are orthogonal projections such…

We study trace functions on the form $ t\to\tr f(A+tB) $ where $ f $ is a real function defined on the positive half-line, and $ A $ and $ B $ are matrices such that $ A $ is positive definite and $ B $ is positive semi-definite. If $ f $…

We prove a matrix trace inequality for completely monotone functions and for Bernstein functions. As special cases we obtain non-trivial trace inequalities for the power function x->x^q, which for certain values of q complement McCarthy's…

Let $J$ denote the interval either $(0,1]$ or $ [1, \infty)$. A positive function $f$ on $J$ with $f(1) =1$ is reffered to as a Weierstrass function if it fulfils the double inequality for $x,y \in J$: $$f(x) + f(y) -1 \leq f(xy) \leq…

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 $\phi$ be a linear map from the $n\times n$ matrices ${\mathcal M}_n$ to the $m\times m$ matrices ${\mathcal M}_m$. It is known that $\phi$ is $2$-positive if and only if for all $K\in {\mathcal M}_n$ and all strictly positive $X\in…

We refine Epstein's method to prove joint concavity/convexity of matrix trace functions of Lieb type $\mathrm{Tr}\,f(\Phi(A^p)^{1/2}\Psi(B^q)\Phi(A^p)^{1/2})$ and symmetric (anti-) norm functions of the form…

We describe the maximal class of functions $f$ on the real line, for which the Lifshitz--Krein trace formula $\operatorname{trace}(f(A)-f(B))=\int_{\Bbb R} f'(s)\boldsymbol{\xi}(s)\,ds$ holds for arbitrary self-adjoint operators $A$ and $B$…

Jensen's trace inequality is established for every multivariable, convex function and every trace or trace-like functional on a C*-algebra.

We use nearly parallel pure states to characterize positive linear functionals $\phi$ on $\mathbb{M}_n$ as positive multiples of the trace if and only if $\phi(A \natural B) \leq \sqrt{\phi(A) \phi(B)}$ for all positive definite matrices…

Some subadditivity results involving symmetric (unitarily invariant) norms are obtained. For instance, if $g(t)=\sum_{k=0}^m a_kt^k$ is a polynomial of degree $m$ with non-negative coefficients, then, for all positive operators $A,\,B$ and…

For any densely defined, lower semi-continuous trace \tau on a C*-algebra A with mutually commuting C*-subalgebras A_1, A_2, ... A_n, and a convex function f of n variables, we give a short proof of the fact that the function (x_1, x_2,…

Let f be a non-negative concave function on the positive half-line. Let A and B be two positive matrices. Then, for all symmetric norms, || f(A+B) || is less than || f(A)+f(B) ||. When f is operator concave, this was proved by Ando and…

We classify functions $f:(a,b)\rightarrow \mathbb{R}$ which satisfy the inequality $$\operatorname{tr} f(A)+f(C)\geq \operatorname{tr} f(B)+f(D)$$ when $A\leq B\leq C$ are self-adjoint matrices, $D= A+C-B$, the so-called trace minmax…

The paper studies the problem, for which continuous functions $f$ on the real line ${\Bbb R}$, the difference of the functions $f(B)-f(A)$ of self-adjoint operators $A$ and $B$ with trace class difference must also be of trace class. The…

In this work, an operator superquadratic function (in operator sense) for positive Hilbert space operators is defined. Several examples with some important properties together with some observations which are related to the operator…

In this note we prove that Tr (MN+ PQ)>= 0 when the following two conditions are met: (i) the matrices M, N, P, Q are structured as follows: M = A -B, N = inv(B)-inv(A), P = C-D, Q =inv (B+D)-inv(A+C), where inv(X) denotes the inverse…