Characterization of decay rates for discrete operator semigroups

TL;DR

This paper studies how quickly the norms of iterated operators decay in discrete semigroups on Hilbert spaces, providing characterizations and estimates that relate decay rates to resolvent growth and integral bounds.

Contribution

It offers new characterizations of decay rates for discrete operator semigroups, linking them to resolvent growth and integral estimates, with applications to perturbed semigroups and Banach space bounds.

Findings

Decay rates characterized by resolvent growth near the spectrum boundary.

Integral estimates provide alternative decay rate characterizations.

Results connect decay behavior with boundedness of certain operator sums.

Abstract

Let $T$ be a power-bounded linear operator on a Hilbert space $X$, and let $S$ be a bounded linear operator from another Hilbert space $Y$ to $X$. We investigate the non-exponential rate of decay of $\|T^nS\|$ as $n \to \infty$. First, when $X = Y$ and $S$ commutes with $T$, we characterize the decay rate of $\|T^nS\|$ in terms of the growth rate of $\|(\lambda I - T)^{-k}S\|$ as $|\lambda| \downarrow 1$ for some $k \in \mathbb{N}$. Next, we provide another characterization by means of an integral estimate of $\|(\lambda I - T)^{-k}S\|$. The second characterization is then applied to asymptotic estimates for perturbed discrete operator semigroups. Finally, we present some results on the relation between the decay rate of $\|T^nS\|$ and the boundedness of the sum $\sum_{n=1}^{\infty} f(n)\|T^nSy\|^p$ for all $y \in Y$ in the Banach space setting, where $f \colon \mathbb{N} \to(0,\infty)$ and $p \geq 1$.