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Counterexamples concerning powers of sectorial operators on a Hilbert space

Published online by Cambridge University Press:  17 April 2009

Arnaud Simard
Arnaud Simard
Affiliation:
Equipe de Mathématiques de BasançonUniversité de Franche-Comté25030 Besancon cedexFrance, e-mail: simard@math.univ-fcomte.fr
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We give explicit constructions of semigroups and operators with particular properties. First we build a bounded C0-semigroup which is invertible and which is not similar to a semigroup of contractions. Afterwards we exhibit operators which admit bounded imaginary powers of angle ω > 0 on a Hilbert space but which do not admit a bounded functional calculus on the sector of angle ω. (This gives the limit of McIntosh's fundamental result.) Finally we build, in the 2-dimensional Hilbert space, an operator which is not the negative generator of a semigroup of contractions, although its imaginary powers are bounded by eπ|s|/2.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 60 , Issue 3 , December 1999 , pp. 459 - 468
Copyright
Copyright © Australian Mathematical Society 1999

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