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Holomorphic functional calculi and sums of commuting opertors

Published online by Cambridge University Press:  17 April 2009

David Albrecht ,
Edwin Franks  and
Alan McIntosh
David Albrecht
Affiliation:
Computer Science Department, Monash University, Claytin Vic 3168, Australia, e-mail: [email protected]
Edwin Franks
Affiliation:
Computer Science Department, Monash University, Claytin Vic 3168, Australia, e-mail: [email protected]
Alan McIntosh
Affiliation:
Department of Mathematics, Macquarie University, New South Wales 2109, Australia, e-mail: [email protected], [email protected]
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Abstract

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Let S and T be commuting operators of type ω and type ϖ in a Banach space X. Then the pair has a joint holomorphic functional calculus in the sense that it is possible to define operators f(S, T) in a consistent manner, when f is a suitable holomorphic function defined on a product of sectors. In particular, this gives a way to define the sum S + T when ω + ϖ < π. We show that this operator is always of type μ where μ = max{ω, ϖ}. We explore when bounds on the individual functional calculi of S and T imply bounds on the functional calculus of the pair (S, T), and some implications for the regularity problem of when ∥(S + T)u∥ is equivalent to ∥Su∥ + ∥Tu∥.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 58 , Issue 2 , October 1998 , pp. 291 - 305
Copyright
Copyright © Australian Mathematical Society 1998

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