A new way of constructing examples in operator ergodic theory

Abstract

We propose a method of constructing examples in operator ergodic theory which unifies and extends some previously known examples. It also allows us to answer several questions that have been open for some time (including a question of Allan [1]).

A note on the powers of Cesàro bounded operators

“…Our main result here is the following theorem which addresses the question presented in [11,Problem 3] of whether it always holds that a Cesàro bounded Hilbert space operator T with a single spectrum {1} satisfies T n+1 − T n → 0 (n → ∞). Our counterexample is a further study of the examples from [11].…”

“…We can construct Cesàro bounded operators from power-bounded operators by building on the following matrix construction that was studied in [11]. Let us choose a power-bounded operator T on X, i.e.…”

“…In the following lemma we summarize some of the basic properties of the T operator, which were described in [11].…”

“…We note that this question originally goes back to Zemánek's paper on the Gelfand-Hille theorems [13]. Here, applying the matrix construction outlined in [11], we will prove that the answer to Zemánek's question is also negative. In our counterexample, convergence holds in the strong operator topology instead of the norm topology, which can be proved by exploiting D. Tsedenbayar's earlier result [12] on the classical Volterra operator.…”

“…One can easily see the necessity of this condition. A negative answer and many interesting examples related to this problem were provided by Tomilov and Zemánek in their paper [11] using an operator matrix construction. However, they left open the following problem: if T is a Cesàro bounded operator in a Hilbert space and its spectrum is just…”