Related papers: Resolvent estimates for operators belonging to exp…

Let $\mathbb{H}^{n}$ be the Heisenberg group. For $0 \leq \alpha < Q=2n+2$ and $N \in \mathbb{N}$ we consider exponent functions $p(\cdot) : \mathbb{H}^{n} \to (0, +\infty)$, which satisfies H\"older conditions, such that $\frac{Q}{Q+N} <…

Let $\lambda$ be a complex number in the closed unit disc $\overline{\Bbb D}$, and $\cal H$ be a separable Hilbert space with the orthonormal basis, say, ${\cal E}=\{e_n:n=0,1,2,\cdots\}$. A bounded operator $T$ on $\cal H$ is called a…

In this paper, we study the operator equation $AB=\lambda BA$ for a bounded operator $A,B$ on a complex Hilbert space. We focus on algebraic relations between different operators that include normal, $M$-hyponormal, quasi $*$-paranormal and…

In this paper, we consider self-adjoint difference equations of the form -\Delta(a_{n-1}\Delta y_{n-1})+b_{n}y_{n}=\lambda y_{n},n=0,1,...\label{eq:abstract} where $a_{n-1}>0$ for all $n\ge0$ and $b_{n}$ are real and $\lambda$ is complex.…

We show that a compact operator $A$ is a multiple of a positive semi-definite operator if and only if $$ \sigma(AB) \subseteq \overline{W(A)W(B)}, \quad\text{for all (rank one) operators $B$}. $$ An example of a normal operator is given to…

Consider the Schroedinger operator with semiclassical parameter h, in the limit where h goes to zero. When the involved long-range potential is smooth, it is well known that the boundary values of the operator's resolvent at a positive…

Several unitarily invariant norm inequalities and numerical radius inequalities for Hilbert space operators are studied. We investigate some necessary and sufficient conditions for the parallelism of two bounded operators. For a finite rank…

Consider $\mathcal{H}$ is a complex Hilbert space and $A$ is a positive operator on $\mathcal{H}.$ The mapping $\langle\cdot,\cdot\rangle_A: \mathcal{H}\times \mathcal{H} \to \mathbb {C}$, defined as $\left\langle…

We address the problem on the right definition of the Schroedinger operator with potential $\delta'$, where $\delta$ is the Dirac delta-function. Namely, we prove the uniform resolvent convergence of a family of Schroedinger operators with…

Using elementary means, we derive the three most popular splittings of $e^{(A+B)}$ and their error bounds in the case when $A$ and $B$ are (possibly unbounded) operators in a Hilbert space, generating strongly continuous semigroups,…

We prove dispersive bounds for fractional Schr\"odinger operators on $\mathbb R^n$ of the form $H=(-\Delta)^{\alpha}+V$ with $V$ a real-valued, decaying potential and $\alpha \notin\mathbb N$. We derive pointwise bounds on the resolvent…

For $\lambda \in (1/2, 1)$ and $\alpha$, we consider sets of numbers $x$ such that for infinitely many $n$, $x$ is $2^{-\alpha n}$-close to some $\sum_{i=1}^n \omega_i \lambda^i$, where $\omega_i \in \{0,1\}$. These sets are in Falconer's…

We study degenerate hypoelliptic Ornstein-Uhlenbeck operators in $L^2$ spaces with respect to invariant measures. The purpose of this article is to show how recent results on general quadratic operators apply to the study of degenerate…

This paper is devoted to the study of a semiclassical "black box" operator $P$. We estimate the norm of its resolvent truncated near the trapped set by the norm of its resolvent truncated on rings far away from the origin. For $z$ in the…

Let $A$ be a positive operator on a complex Hilbert space $\mathcal{H}.$ We present inequalities concerning upper and lower bounds for $A$-numerical radius of operators, which improve on and generalize the existing ones, studied recently in…

In this paper we continue the study on the resolvent estimates of the Laplace-Beltrami operator $\Delta_g$ on a compact manifolds $M$ with dimension $n\geq3$. On the Sobolev line $1/p-1/q=2/n$ we can prove that the resolvent…

This article studies sharp norm estimates for the commutator of pseudo-differential operators with multiplication operators on closed Heisenberg manifolds. In particular, we obtain a Calderon commutator estimate: If $D$ is a first-order…

We study the acoustic operator $A_{v,\rho }:=v^{2}\rho\nabla\!\cdot\rho^{-1}\nabla$ with transmission conditions at the boundary of $\Omega=\Omega_{1}\cup\dots\cup\Omega_{n}$, where the $\Omega_{\ell}$'s are connected disjoint open bounded…

Given a Hilbert space and the generator $A$ of a strongly continuous, exponentially stable, semigroup on this Hilbert space. For any $g(-s) \in {\mathcal H}_{\infty}$ we show that there exists an infinite-time admissible output operator…

This paper introduces a new definition of $\alpha$-monotone operators in real 2-uniformly convex and smooth Banach spaces. Based on this new definition, we establish several novel structural and analytical properties of such operators,…