Related papers: Lipschitz Estimates and an application to trace fo…

We employ techniques from optimal transport in order to prove decay of transfer operators associated to iterated functions systems and expanding maps, giving rise to a new proof without requiring a Doeblin-Fortet (or Lasota-Yorke)…

Given $H$ self-adjoint, $V$ symmetric and relatively $H$-bounded, and $f:\mathbb{R}\to\mathbb{C}$ satisfying mild conditions, we show that the Gateaux derivative $$\frac{d^n}{dt^n}f(H+tV)|_{t=0}$$ exists in the operator norm topology, for…

We revisit and prove some convexity inequalities for trace functions conjectured in the earlier part I. The main functional considered is \Phi_{p,q}(A_1,A_2,...,A_m) = (trace((\sum_{j=1}^m A_j^p)^{q/p}))^{1/q} for m positive definite…

The operator function (A,B)\to\tr f(A,B)(K^*)K, defined on pairs of bounded self-adjoint operators in the domain of a function f of two real variables, is convex for every Hilbert Schmidt operator K, if and only if f is operator convex. As…

We survey the notion of the spectral shift function of a pair of self-adjoint operators and recent progress on its connection with the Witten index. We also describe a proof of Krein's Trace Theorem that does not use complex analysis [53]…

We show how a matrix version of the Buslaev-Faddeev-Zakharov trace formulae for a one-dimensional Schr\"odinger operator leads to Lieb-Thirring inequalities with sharp constants $L^{cl}_{\gamma,d}$ with $\gamma\ge 3/2$ and arbitrary $d\ge…

In this paper we introduce and study pseudo-differential operators with operator valued symbols on the abstract Heisenberg group $\mathbb{H}(G):=G \times \widehat{G} \times \mathbb{T},$ where $G$ a locally compact abelian group with its…

We propose a method for solving constrained fixed point problems involving compositions of Lipschitz pseudo contractive and firmly nonexpansive operators in Hilbert spaces. Each iteration of the method uses separate evaluations of these…

The paper establishes the Krein and Koplienko trace formulas for multivariable operator functions on symmetrically normed ideals of bounded operators. Results are proved for self-adjoint and maximal dissipative operators. They cover both…

We construct higher order spectral shift functions, extending the perturbation theory results of M. G. Krein and L. S. Koplienko on representations for the remainders of the first and second order Taylor-type approximations of operator…

A $1$-Lipschitz map $f$ from a convex compact set to itself has fixed points. This consequence of Brouwer's or Schauder's fixed point theorem has more elementary proofs by approximating $f$ by $\lambda$-contractions, $f_\lambda$. We study…

The introduction of abstract Friedrichs operators in 2007-an operator-theoretic framework for studying classical Friedrichs operators has led to significant developments in the field, including results on well-posedness, multiplicity, and…

Let $H_0 = -\Delta + V_0(x)$ be a Schroedinger operator on $L_2(\mathbb{R}^\nu),$ $\nu=1,2,$ or 3, where $V_0(x)$ is a bounded measurable real-valued function on $\mathbb{R}^\nu.$ Let $V$ be an operator of multiplication by a bounded…

In this paper we study the joint convexity/concavity of the trace functions \[ \Psi_{p,q,s}(A,B)=\text{Tr}(B^{\frac{q}{2}}K^*A^{p}KB^{\frac{q}{2}})^s,~~p,q,s\in \mathbb{R}, \] where $A$ and $B$ are positive definite matrices and $K$ is any…

Sharp Strichartz estimates are proved for Schr\"odinger and wave equations with Lipschitz coefficients satisfying additional structural assumptions. We use Phillips functional calculus as a substitute for Fourier inversion, which shows how…

Let $p>2$ be a prime number. In this short note, we calculate explicitly the unitary dual and the matrix coefficients of the Heisenberg group over the $p$-adic integers. As an application, we consider directional Vladimirov-Taibleson…

Let $\Op_t(a)$, for $t\in \mathbf R$, be the pseudo-differential operator $$ f(x) \mapsto (2\pi)^{-n}\iint a((1-t)x+ty,\xi)f(y)e^{i\scal {x-y}\xi} dyd\xi $$ and let $\mathscr I_p$ be the set of Schatten-von Neumann operators of order $p\in…

We establish the following results on higher order $\mathcal{S}^p$-differentiability, $1<p<\infty$, of the operator function arising from a continuous scalar function $f$ and self-adjoint operators defined on a fixed separable Hilbert…

In the recent paper, \cite{tilli} Nicola and Tilli proved the Faber-Krahn inequality, which for $p=2$, states the following. If $f\in\mathcal{F}_\alpha^2$ is an entire function from the corresponding Fock space, then…

We consider harmonic Toeplitz operators $T_V = PV:{\mathcal H}(\Omega) \to {\mathcal H}(\Omega)$ where $P: L^2(\Omega) \to {\mathcal H}(\Omega)$ is the orthogonal projection onto ${\mathcal H}(\Omega) = \left\{u \in L^2(\Omega)\,|\,\Delta u…