Functional models up to similarity and a-contractions

Abstract

We study the generalization of m-isometries and m-contractions (for positive integers m) to what we call a-isometries and a-contractions for positive real numbers a. We show that an operator satisfying a certain inequality in hereditary form is similar to a-contraction. This improvement of [9, Theorem I] is based on some Banach algebras techniques. We show that our operator classes are closely connected with fractional finite differences. Using this techniques, we get that, given \(0<b<a\), an a-contraction need not to be a b-contraction in general, but is a b-contraction if a natural additional requirement is imposed.