Derivative bounded functional calculus of power bounded operators on Banach spaces

Abstract

In this article we study bounded operators T on a Banach space X which satisfy the discrete Gomilko-Shi-Feng condition integral(2 pi)(0) vertical bar < R(re(it), T)(2)x, x*>vertical bar dt <= C/(r(2) - 1) parallel to x parallel to parallel to x*parallel to, r > 1, x is an element of X, x* is an element of 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 spaces the discrete Gomilko-Shi-Feng condition is equivalent to power-boundedness. Finally we discuss the last equivalence on general Banach spaces involving the concept of gamma-boundedness.