Related papers: Convex Multivariable Trace Functions

All continuous, SL$(n)$ and translation invariant valuations on the space of convex functions on ${\mathbb R}^n$ are completely classified.

A real valued function $f$ defined on a convex $K$ is anemconvex function iff it satisfies $$ f((x+y)/2) \le (f(x)+f(y))/2 + 1. $$ A thorough study of approximately convex functions is made. The principal results are a sharp universal upper…

The eigenvalues of a self-adjoint nxn matrix A can be put into a decreasing sequence $\lambda=(\lambda_1,...,\lambda_n)$, with repetitions according to multiplicity, and the diagonal of A is a point of $R^n$ that bears some relation to…

This study focuses on convex functions and their generalized. Thus, we start this study by giving the definition of convex functions and some of their properties and discussing a simple geometric property. Then we generalize E-convex…

In the dual $L_{\Phi^*}$ of a $\Delta_2$-Orlicz space $L_\Phi$, that we call a dual Orlicz space, we show that a proper (resp. finite) convex function is lower semicontinuous (resp. continuous) for the Mackey topology…

We define convexity canonically in the setting of monoids. We show that many classical results from convex analysis hold for functions defined on such groups and semigroups, rather than only on vector spaces. Some examples and…

The trace on matrix rings, along with the augmentation map and Kaplansky trace on group rings, are some of the many examples of linear functions on algebras that vanish on all commutators. We generalize and unify these examples by studying…

We present some results on the monotonicity of some traces involving functions of self-adjoint operators with respect to the natural ordering of their associated quadratic forms. We also apply these results to complete a proof of the Wegner…

In this note we study restrictions on the recently introduced super-additive and sub-additive transformations, $A\mapsto A^*$ and $A\mapsto A_*$, of an aggregation function $A$. We prove that if $A^*$ has a slightly stronger property of…

Let $A$ be a positive definite operator on a Hilbert space $H$, and $|||.|||$ be a unitarily invariant norm on $B(H)$. We show that if $f$ is an operator monotone function on $(0,\infty)$ and $n\in \mathbb{N}$, then $|||D^n…

Caffarelli's contraction theorem bounds the derivative of the optimal transport map between a log-convex measure and a strongly log-concave measure. We show that an analogous phenomenon holds on the level of the trace: The trace of the…

We establish the invariant trace formula (\`a la Arthur) for the ad\'elic covers of connected reductive groups over a number field, under the hypothesis that the trace Paley-Wiener theorem is verified for all Levi subgroups at the real…

The polynomial relationship between elementary symmetric functions (Cauchy enumeration formula) is formulated via a ``raising operator" and Fock space construction. A simple graphical proof of this relation is proposed. The new operator…

In the present paper, we first prove the logarithmic convexity of the elementary function $\frac{b^x-a^x}x$, where $x\ne0$ and $b>a>0$. Basing on this, we then provide a simple proof for Schur-convex properties of the extended mean values,…

We consider nonconvex real valued functions whose truncations are either quasiconvex or even convex starting with a certain level. Among them, the $C^2$-smooth functions whose level sets are all completely contained in the positive definite…

We study convexity or concavity of certain trace functions for the deformed logarithmic and exponential functions, and obtain in this way new trace inequalities for deformed exponentials that may be considered as generalizations of…

We consider the operator algebra $\mathscr A$ on $\mathscr S(\mathbb R^n)$ generated by the Shubin type pseudodifferential operators, the Heisenberg-Weyl operators and the lifts of the unitary operators on $\mathbb C^n$ to metaplectic…

A differentiable function is pseudoconvex if and only if its restrictions over straight lines are pseudoconvex. A differentiable function depending on one variable, defined on some closed interval $[a,b]$ is pseudoconvex if and only if…

For all the convolution algebras $L^1[0,1),\ L^1_{\text{loc}}$ and $A(\omega)=\bigcap_n L^1(\omega_n)$, the derivations are of the form $D_{\mu} f=Xf*\mu$ for suitable measures $\mu$, where $(Xf)(t)=tf(t)$. We describe the (weakly) compact…

We consider locally equi-continuous strongly continuous semigroups on locally convex spaces (X,tau). First, we show that if (X,tau) has the property that weak* compact sets of the dual are equi-continuous, then strong continuity of the…