Characterization of decay rates for discrete operator semigroups

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 $p \geq 1$.