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Banach space operators with a bounded H∞ functional calculus

Part of: General theory of linear operators

Published online by Cambridge University Press:  09 April 2009

Michael Cowling ,
Ian Doust ,
Alan Micintosh  and
Atsushi Yagi
Michael Cowling
Affiliation:
School of MathematicsUniversity of New South WalesSydney NSW 2052Australia e-mail: m.cowling@unsw.edu.au
Ian Doust
Affiliation:
School of MathematicsUniversity of New South WalesSudney NSW 2052Australia e-mail: i.doust@unsw.edu.au
Alan Micintosh
Affiliation:
School of Mathematics, Physics, Computing and ElectronicsMacquire UniversityNSW 2109Australia e-mail: alan@macadam.mpce.mq.edu.au
Atsushi Yagi
Affiliation:
Department of MathematicsHimeji Institute of Technology2167 Shosha, Himeji Hyogo 671-22, Japan
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Abstract

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In this paper, we give a general definition for f(T) when T is a linear operator acting in a Banach space, whose spectrum lies within some sector, and which satisfies certain resolvent bounds, and when f is holomorphic on a larger sector.

We also examine how certain properties of this functional calculus, such as the existence of a bounded H∈ functional calculus, bounds on the imaginary powers, and square function estimates are related. In particular we show that, if T is acting in a reflexive Lp space, then T has a bounded H∈ functional calculus if and only if both T and its dual satisfy square function estimates. Examples are given to show that some of the theorems that hold for operators in a Hilbert space do not extend to the general Banach space setting.

MSC classification

Secondary: 47A60: Functional calculus
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 60 , Issue 1 , February 1996 , pp. 51 - 89
Copyright
Copyright © Australian Mathematical Society 1996

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