Perturbations of strongly continuous operator semigroups, and matrix Muckenhoupt weights

Abstract

Abstract. Let A and A 0 be linear continuously invertible operators on a Hilbert space H such that A −1 − A −1 0 has finite rank. Assuming that σ(A 0 ) = ∅ and that the operator semigroup V + (t) = exp{iA 0 t}, t 0, is of class C 0 , we state criteria under which the semigroups U ± (t) = exp{±iAt}, t 0, are of class C 0 as well. The analysis in the paper is based on functional models for nonself-adjoint operators and techniques of matrix Muckenhoupt weights.Key words: nonself-adjoint operator, perturbation of a semigroup, functional model, Muckenhoupt condition. To the centenary of M. G. KreinIn this paper, we apply functional models of nonself-adjoint operators and the technique of matrix Muckenhoupt weights to the theory of one-parameter operator semigroups in Hilbert spaces.1. w-perturbations of linear operators. Let A 0 and A be linear unbounded densely defined and continuously invertible operators on a Hilbert space H such thatwhere f k , g k ∈ H, 1 k n, and the parentheses stand for the inner product in H. We shall assume that the operator A 0 belongs to the class Σ (exp) , that is, and {g k } n 1 such that the operator A related to A 0 ∈ Σ (exp) by (1) generates a C 0 -semigroup U + (t) := exp{iAt} or a C 0 -semigroup U − (t) := exp{−iAt}, t 0. In this paper, we indicate a class of finite-dimensional perturbations A satisfying (1) in which this problem admits a solution.An n × n matrix weight almost everywhere positive on the real line will be denoted by w 2 (x), x ∈ R. Further, by M 2 n we denote the class of matrix weights w 2 satisfying Muckenhoupt's conditionwhere M (w ±2 ) := |∆| −1 ∆ w ±2 (x) dx, ∆ is an arbitrary interval in R, and |∆| is its length. Using the operator A 0 ∈ Σ (exp) and the vectors {g k } n 1 , we construct the n-dimensional rowand introduce the following definition. Definition 1. Suppose that the operators A and A 0 are related by (1).We say that A is a w-perturbation of rank n of the operator A 0 ∈ Σ (exp) if there exists a weight w 2 (x) of the class M 2 n such that the following conditions hold: 1. For each h ∈ H, the function A 0 (x, h)w −1 (x) belongs to L 2 (R).

A criterion for the unconditional basis property of eigenvectors for finite-rank perturbations of Volterra operators

“…The proof of Theorem 2 is based on the development of the approach to finite rank perturbations of Volterra operators presented in [1] * . The definition of a w-perturbation and the correspondence between such perturbations and pairs w 2 , Θ (see Theorem 1) is borrowed from [1].…”

“…The definition of a w-perturbation and the correspondence between such perturbations and pairs w 2 , Θ (see Theorem 1) is borrowed from [1].…”

“…In this paper, which is a continuation of [1], we study finite-rank perturbations K of operators B ∈ Ω (exp) , i.e., operators defined by…”

Функциональные Модели Несамосопряженных Операторов, Сильно Непрерывные Полугруппы И Матричные Веса Макенхаупта

“…Настоящая статья, как и ее краткое изложение [11], посвящается столетию М. Г. Крейна; воспоминания о годах, проведенных рядом с ним, навсегда останутся в нашей памяти.…”