We present a scenario concerning the existence of a large class of reflectionless selfadjoint analytic difference operators. In order to exemplify this scenario, we summarize our results on reflectionless self-adjoint difference operators of relativistic CalogeroMoser type.

The aim of this paper is to investigate spectral properties of second order elliptic operators with measurable coefficients. Namely, we study the problems of L-independence of the spectrum and stability of the essential spectrum. The problem of L-independence of the spectrum for elliptic operators has a long history going back to B. Simon [30] where the question was posed for Schrödinger operat...

Some vector and operator generalized trapezoidal inequalities for continuous functions of selfadjoint operators in Hilbert spaces are given. Applications for power and logarithmic functions of operators are provided as well.

Some inequalities for convex functions of selfadjoint operators in Hilbert spaces under suitable assumptions for the involved operators are given. Applications for particular cases of interest are also provided.

We study degenerate hypoelliptic Ornstein-Uhlenbeck operators in L 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 hypoelliptic Ornstein-Uhlenbeck operators. We first show that some known results about the spectral and subelliptic properties of Ornstein-Uhlenbeck operators may be...

Inequalities of Jackson and Bernstein type are derived for polynomial approximation on simplices with respect to Sobolev norms. Although we do not find simple bases when looking at 120 years of research of orthogonal polynomials on triangles, sharp estimates are obtained from a decomposition into orthogonal subspaces. The formulas reflect the symmetries of simplices, but analogous estimates on ...

The multivariate orthogonal polynomials are related to a family of operators whose matrix representations are block Jacobi matrices. A sufficient condition is given so that these operators, in general unbounded, are commuting and selfadjoint. The spectral theorem for these operators is used to establish the existence of the measure of orthogonality in Favard's theorem.

We prove a general operator theoretic result that asserts that many multiplicity two selfadjoint operators have simple singular spectrum. © 2005 Elsevier Inc. All rights reserved.