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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: [email protected]
Ian Doust
Affiliation:
School of MathematicsUniversity of New South WalesSudney NSW 2052Australia e-mail: [email protected]
Alan Micintosh
Affiliation:
School of Mathematics, Physics, Computing and ElectronicsMacquire UniversityNSW 2109Australia e-mail: [email protected]
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

References

[1]Baillon, J. B. and Clément, Ph., ‘Examples of unbounded imaginary powers of operators’, J. Funct. Anal. 100 (1991), 419–434.CrossRefGoogle Scholar
[2]Boyadzhiev, K. and de Laubenfels, R., ‘Semigroups and resolvents of bounded variation, imaginary powers and H ∞ functional calculus’, Semigroup Forum 45 (1992), 372–384.CrossRefGoogle Scholar
[3]Coifman, R. R., Rochberg, R. and Weiss, G., ‘Applications of transference: the Lρ version of von Neumann's inequality and Littlewood-Paley-Stein theory’, in: Linear spaces and approximation (eds. Butzer, P. L. and Sz.-Nagy, B.) (Birkhäuser, Basel, 1978) pp. 53–67.CrossRefGoogle Scholar
[4]Cowling, M., ‘Harmonic analysis on semigroups’, Ann. of Math. 117 (1983), 267–283.CrossRefGoogle Scholar
[5]Dore, G. and Venni, A., ‘On the closedness of the sum of two closed operators’, Math. Z. 196 (1987), 189–201.CrossRefGoogle Scholar
[6]Dunford, N. and Schwartz, J. T., Linear operators, Part I: General theory (Wiley Interscience, New York, 1958).Google Scholar
[7]Duong, X. T., ‘From the L 1 norms of the complex heat kernels to a Hörmander multiplier theorem for sub-Laplacians on nilpotent lie groups’, University of NSW Math. Report (1993).Google Scholar
[8]Duong, X. T. and McIntosh, A., ‘Functional calculi of second order elliptic operators with bounded measurable coefficients’, Macquarie Math. Report 92–119 (1992).Google Scholar
[9]Edwards, R. E. and Gaudry, G. I., Littlewood-Paley and multiplier theory (Springer, Berlin, 1977).CrossRefGoogle Scholar
[10]Giga, Y. and Sohr, H., ‘Abstract Lp estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains’, J. Funct. Anal. 102 (1991), 72–94.CrossRefGoogle Scholar
[11]Kato, T., ‘Remarks on pseudo-resolvents and infinitesimal generators of semigroups’, Proc. Japan Acad. Ser. A Math. Sci. 35 (1959), 467–468.CrossRefGoogle Scholar
[12]Kato, T., Perturbation theory of linear operators (Springer, New York, 1966).Google Scholar
[13]Komatsu, H., ‘An ergodic theorem’, Proc. Japan Acad. Ser. A Math. Sci. 44 (1968), 46–48.CrossRefGoogle Scholar
[14]de Leeuw, K., ‘On Lp multipliers’, Ann. of Math. 81 (1965), 364–379.CrossRefGoogle Scholar
[15]McIntosh, A., ‘Operators which have an H∞ functional calculus’, Miniconference on operator theory and partial differential equations (Proc. Centre Math. Analysis, 14, A.N.U., Canberra, 1986), pp. 210–231.Google Scholar
[16]McIntosh, A. and Yagi, A., ‘Operators of type w without a bounded H∞ functional calculus’, Miniconference on operators in analysis (Proc. Centre Math. Analysis, 24, A.N.U., Canberra, 1989), pp. 159–172.Google Scholar
[17]Prüss, J. and Sohr, H., ‘On operators with bounded imaginary powers in Banach spaces’, Math. Z. 203 (1990), 429–452.CrossRefGoogle Scholar
[18]Seeley, R., ‘Norms and domains of the complex powers ABz’, Amer. J. Math. 93 (1971), 299–309.CrossRefGoogle Scholar
[19]Stein, E. M., Topics in harmonic analysis related to the Littlewood-Paley theory, Annals of Math. Studies 63 (Princeton University Press, Princeton, N. J., 1970).CrossRefGoogle Scholar
[20]Titchmarsh, E. C., The theory of functions (2nd ed.) (Oxford University Press, London, 1939).Google Scholar
[21]Yagi, A., ‘Coïncidence entre des espaces d'interpolation et des domaines de puissances fractionaries d'opérateurs’, C. R. Acad. Sci. Paris (Sér. I) 299 (1984), 173–176.Google Scholar
[22]Yosida, K., Functional analysis (6th ed.) (Springer, Berlin, 1980).Google Scholar
[23]Zygmund, A., Trigonometric series, vol. I, (2nd revised edition) (Cambridge University Press, Cambridge, 1968).Google Scholar