Subordinated discrete semigroups of operators

TL;DR

This paper investigates the asymptotic behavior of subordinated discrete semigroups of operators formed by weighted sums of powers of a power-bounded operator, focusing on conditions ensuring specific resolvent properties.

Contribution

It introduces new conditions on the probability measure F that guarantee the subordinated operator S satisfies the Ritt resolvent condition, expanding understanding of operator semigroup asymptotics.

Findings

Identification of conditions on F for Ritt property of S

Examples illustrating when the property holds or fails

Extension of results to Kreiss resolvent condition

Abstract

Given a power-bounded linear operator T in a Banach space and a probability F on the non-negative integers, one can form a `subordinated' operator S = \sum_k F(k) T^k. We obtain asymptotic properties of the subordinated discrete semigroup (S^n: n=1,2,...) under certain conditions on F. In particular, we study probabilities F with the property that S satisfies the Ritt resolvent condition whenever T is power-bounded. Examples and counterexamples of this property are discussed. The hypothesis of power-boundedness of T can sometimes be replaced by the weaker Kreiss resolvent condition.