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Modified cauchy kernels and functional calculus for operators on Banach space

Part of: General theory of linear operators

Published online by Cambridge University Press:  09 April 2009

Edwin Franks
Edwin Franks
Affiliation:
Depart of Mathematics Macquarie UniversityNSW 2109Australia e-mail: edwin@mpce.mq.edu.au
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Abstract

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In Banach space operators with a bounded H∞ functional calculus, Cowling et al. provide some necessary and sufficient conditions for a type-ω operator to have a bounded H∞ functional calculus. We provide an alternate development of some of their ideas using a modified Cauchy kernel which is L1 with respect to the measure ]dz]/]z]. The method is direct and has the advantage that no transforms of the functions are necessary.

Keywords

functional calculusmatricial functional calculus.

MSC classification

Secondary: 47A60: Functional calculus
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 63 , Issue 1 , August 1997 , pp. 91 - 99
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Cowling, M., Doust, I., McIntosh, A., and Yagi, A., ‘Banach space operators with a bounded H∞ functional calculus’, J. Austral. Math. Soc. (Series A) 60 (1996), 51–89.CrossRefGoogle Scholar
[2]Foguel, S. R., ‘A counterexample to a problem of Sz.-Nagy’, Proc. Amer. Math. Soc. 15 (1964), 788–790.CrossRefGoogle Scholar
[3]Franks, E. and McIntosh, A., ‘Discrete quadratic estimates and holomorphic functional calculi of operators in Banach spaces’, preprint (1995).Google Scholar
[4]Franks, E. and Ryan, J., ‘Bounded monogenic functions on unbounded domains’, preprint (1995).Google Scholar
[5]Halmos, P. R., ‘On Foguel's answer to Nagy's problem’, Proc. Amer. Math. Soc. 15 (1964), 791–793.Google Scholar
[6]McIntosh, A., ‘Operators which have an H∞ functional calculus’, in: Miniconference on Operator Theory and Partial Differential Equations, Proc. Centre Math. Analysis, A.N.U., Canberra 14 (1986), 210–231.Google Scholar
[7]McIntosh, A. and Yagi, A., ‘Operators of type-ω without a bounded H ∞ functional calculus’, in: Miniconference on Operators in Analysis, Proc. Centre. Math. Analysis, A.N.U., Canberra 24 (1989), 159–172.Google Scholar
[8]Le Merdy, C., ‘The similarity problem for bounded analytic semigroups on Hilbert space’, preprint (1995).Google Scholar
[9]Paulsen, V. I., ‘Every completely polynomially bounded operator is similar to a contraction’, J. Funct. Anal. 55 (1984), 1–17.CrossRefGoogle Scholar