A note on powers of a contraction
by
Zoltán Léka
University of Szeged, Hungary

In this talk we shall prove the following: Let T be a contraction acting on a Hilbert space H such that s(T) ∩{z ∈ C : |z| = 1 } ≠ ∅. Let us choose a bounded, linear operator Q on H which commutes with T. Then, for every l ∈ s(T) ∩{z ∈ C : |z| = 1 },

lim n → ∞  1 n n å k=1  l-k Tk Q = 0

holds, if and only if lim_{n → ∞} TⁿQ = 0.

We shall also discuss the connection of this result to the Katznelson-Tzafriri theorem and its extensions in a Hilbert space setting. The latter (i.e. the extensions) have been proved by J. Esterle, E. Strouse & F. Zouakia, and H. Bercovici.

Date received: June 11, 2008