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Keywords:
bounded linear operator; uniform ergodic theorem; Nörlund means of operator iterates; spectrum; pole of the resolvent
bounded linear operator; uniform ergodic theorem; Nörlund means of operator iterates; spectrum; pole of the resolvent
Summary:
We are concerned here with relating the spectral properties of a bounded linear operator $T$ on a Banach space to the behaviour of the means $(1/{s(n)})\sum _{k=0}^n(\Delta s)(n-k)T^k$, where $s$ is a nondecreasing sequence of positive real numbers, and $\Delta $ denotes the inverse of the automorphism on the vector space of scalar sequences which maps each sequence into the sequence of its partial sums. In a previous paper, we obtained a uniform ergodic theorem for the means above, under the hypotheses $\lim _{n\rightarrow \infty }s(n)=\infty$, $\lim _{n\rightarrow \infty }{s(n+1)}/{s(n)}=1$, and $\Delta ^qs\in \ell _1$ for a positive integer $q$: indeed, we proved that if $T^n/s(n)$ converges to zero in the uniform operator topology for such a sequence $s$, then the averages above converge in the same topology if and only if 1 is either in the resolvent set of $T$, or a simple pole of the resolvent function of $T$. In this paper, we prove that if $\liminf _{n\rightarrow \infty }{s(n+1)}/{s(n)}=1$, and the averages above converge in the uniform operator topology, then 1 is either in the resolvent set of $T$, or a simple pole of the resolvent function of $T$. The converse is not true, even if the sequence $s$ satisfies all the hypotheses of the theorem recalled above, except membership of $\Delta ^qs$ in $\ell _1$ for a positive integer $q$. We also prove that if $\lim _{n\rightarrow \infty }\root n\of {s(n)}=1$, and the function $h_s(z)=\sum _{n=0}^{\infty }s(n)z^n$ has no zeros in the open unit disk, then operator norm boundedness of the averages of the sequence $T^n$induced by $s$ implies that the spectral radius of $T$ is less than or equal to $1$. This result fails if the assumption about $h_s$ is dropped. Indeed, it may happen that the averages converge in the uniform operator topology for a sequence $s$ satisfying $\lim _ {n\rightarrow \infty }s(n)=\infty $, $\lim _{n\rightarrow \infty } {s(n+1)}/{s(n)}=1$, and $\Delta ^qs\in l_1$ for a positive integer $q$, and nevertheless the spectral radius of $T$ is strictly larger than 1.
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