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

Abstract

A criterion for the unconditional basis property of eigenvectors for finite-rank perturbations of Volterra operators is given. Considerations are based on functional models for non-selfadjoint operators and on the technique of the Muckenhoupt matrix weights. Key words: unconditional basis, non-self-adjoint operators, entire inner matrix functions, Muckenhoupt matrix weights. This work is dedicated to the outstanding mathematician B. S. Pavlov on the occasion of his birthday. Pavlov has made an important contribution to the theory of unconditional bases formed by eigenvectors for various classes of non-self-adjoint operators, as well as to other areas of mathematics. 1. Let Ω (exp) denote the class of operators B on a separable Hilbert space H which satisfy the following conditions: (i) σ(B) = {0} and Ker B = Ker B * = {0}; (ii) the operator B −1 generates a C 0 -semigroup exp{iB −1 t}, t > 0; (iii) the entire operator function (I−zB) −1 is of finite exponential growth.Note that antidissipative operators B (i.e., operators B for which (B − B * )/i 0) satisfying conditions (i) and (iii) and all operators similar to them belong to Ω (exp) .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