On polynomially bounded operators acting on a Banach space

Abstract

By the Von Neumann inequality every contraction on a Hilbert space is polynomially bounded. A simple example shows that this result does not extend to Banach space contractions. In this paper, we give general conditions under which an arbitrary Banach space contraction is polynomially bounded. These conditions concern the thinness of the spectrum and the behaviour of the resolvent or the sequence of negative powers. To do this we use techniques from harmonic analysis, in particular, results concerning thin sets such as Helson sets, Kronecker sets and sets that satisfy spectral synthesis.

Asymptotic behavior of polynomially bounded operators

“…The similar result holds for polynomially bounded operators [3] (for related results see also [2,5,8,10]). We see that under the assumptions of Esterle-Strouse-Zouakia Theorem the Lebesgue measure of σ u (T ) is necessarily zero.…”

Reflexive Banach spaces with all power-bounded operators almost periodic

“…Zarrabi [26,Theorem 3.1] proved that if an invertible contraction (power-bounded) T on a complex Banach space has countable spectrum σ(T ) ⊂ T and log T −n √ n → 0, then T is an isometry (T −1 is power-bounded). If, in addition, the countable spectrum is a Helson subset of T, then T is polynomially bounded [27,Theorem 4.2].…”

Some results of Katznelson–Tzafriri type

“…We shall now denote by T a representation of S by contractions on a Banach space. We show (Corollary 5.5) that equality (1.2) holds for every T such that Sp u (T , S) = E, if and only if E satisfies spectral synthesis and is a Helson set with α(E) = 1, where α(E) is the Helson constant of E. See [10] for further results on Helson sets that satisfy spectral synthesis.…”