Characterization of decay rates for discrete operator semigroups

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 $\Vert T^{n}S\Vert$ as $n\to \infty$. First, when $X=Y$ and $S$ commutes with $T$, we characterize the decay rate of $\Vert T^{n}S\Vert$ in terms of the growth rate of $\Vert (\hat{I}»I-T{)}^{-k}S\Vert$ as $|\hat{I}»|\downarrow 1$ for some $k\in \mathbb{N}$. Next, we provide another characterization by means of an integral estimate of $\Vert (\hat{I}»I-T{)}^{-k}S\Vert$. 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 $\Vert T^{n}S\Vert$ and the boundedness of the sum ${\sum }_{n=1}^{\infty }f(n)\Vert T^{n}Sy{\Vert }^p$ for all $y\in Y$ in the Banach space setting, where $f:\mathbb{N}\to (0,\infty )$ and $p\ge 1$.

28 pages. To appear in Studia Mathematica