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REMARKS ON ℓ1 AND -MAXIMAL REGULARITY FOR POWER-BOUNDED OPERATORS

Part of: General theory of linear operators Difference equations Groups and semigroups of linear operators, their generalizations and applications Difference and functional equations

Published online by Cambridge University Press:  01 June 2008

N. J. KALTON  and
P. PORTAL
N. J. KALTON
Affiliation:
Department of Mathematics, University of Missouri–Columbia, Columbia, MO 65211, USA (email: nigel@math.missouri.edu)
P. PORTAL*
Affiliation:
Centre for Mathematics and its Applications, Australian National University, Canberra, ACT 0200, Australia (email: pierre.portal@maths.anu.edu.au)
*
✉For correspondence; e-mail: pierre.portal@maths.anu.edu.au
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Abstract

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We discuss ℓp-maximal regularity of power-bounded operators and relate the discrete to the continuous time problem for analytic semigroups. We give a complete characterization of operators with ℓ1 and -maximal regularity. We also introduce an unconditional form of Ritt’s condition for power-bounded operators, which plays the role of the existence of an -calculus, and give a complete characterization of this condition in the case of Banach spaces which are L1-spaces, C(K)-spaces or Hilbert spaces.

Keywords

power-bounded operatorsRitt’s conditionmaximal regularityH inftysquare function

MSC classification

Secondary: 47A30: Norms (inequalities, more than one norm, etc.) 47A60: Functional calculus 47D06: One-parameter semigroups and linear evolution equations 39A12: Discrete version of topics in analysis
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 84 , Issue 3 , June 2008 , pp. 345 - 365
Copyright
Copyright © 2008 Australian Mathematical Society

Footnotes

The first author acknowledges support from NSF grant DMS-0244515.

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