Derivative bounded functional calculus of power bounded operators on Banach spaces

TL;DR

This paper investigates a specific boundedness condition for operators on Banach spaces, establishing its equivalence to various functional calculi and exploring its implications for power-boundedness and $\,\gamma$-boundedness.

Contribution

It introduces the discrete Gomilko Shi-Feng condition and proves its equivalence to derivative bounded and Besov space functional calculi, extending results to Hilbert and general Banach spaces.

Findings

Gomilko Shi-Feng condition is equivalent to a derivative bounded functional calculus.

On Hilbert spaces, this condition is equivalent to power-boundedness.

The paper discusses $\,\gamma$-boundedness in the context of Banach spaces.

Abstract

In this article we study bounded operators $T$ on Banach space $X$ which satisfy the discrete Gomilko Shi-Feng condition $$\int_{0}^{2\pi}|\langle R(re^{it},T)^{2}x,x^*\rangle |dt \leq \frac{C}{(r^2-1)}\norme{x}\norme{x^*},\quad r>1, x\in X, x^* \in X^*. $$ We show that it is equivalent to a certain derivative bounded functional calculus and also to a bounded functional calculus relative to Besov space. Also on Hilbert space discrete Gomilko Shi-Feng condition is equivalent to power-boundedness. Finally we discuss the last equivalence on general Banach space involving the concept of $\gamma$-boundedness.