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Operator-valued multiplier theorems characterizing Hilbert spaces

Part of: Nontrigonometric harmonic analysis Harmonic analysis in one variable Normed linear spaces and Banach spaces; Banach lattices General theory of linear operators

Published online by Cambridge University Press:  09 April 2009

Wolfgang Arendt  and
Shangquan Bu
Wolfgang Arendt
Affiliation:
Abteilung Angewandte Analysis, Universität Ulm, 89069 Ulm, Germany e-mail: arendt@mathematik. uni-ulm.de
Shangquan Bu
Affiliation:
Department of Mathematical Science, University of Tsinghua, Beijing 100084, China e-mail: [email protected]
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Abstract

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We show that the operator-valued Marcinkiewicz and Mikhlin Fourier multiplier theorem are valid if and only if the underlying Banach space is isomorphic to a Hilbert space.

Keywords

Operator-valued Fourier multiplierHilbert space and Rademacher boundedness

MSC classification

Secondary: 42A45: Multipliers 42C99: None of the above, but in this section 46C15: Characterizations of Hilbert spaces 47A56: Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 77 , Issue 2 , October 2004 , pp. 175 - 184
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Amann, H., ‘Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications’, Math. Nachr. 186 (1997), 5–56.Google Scholar
[2]Arendt, W., Batty, C. and Bu, S., ‘Fourier multipliers for Hölder continuous functions and maximal regularity’, Studia Math. 160 (2004), 23–51.CrossRefGoogle Scholar
[3]Arendt, W. and Bu, S., ‘The operator-valued Marcinkiewicz multiplier theorem and maximal regularity’, Math. Z. 240 (2002), 311–343.CrossRefGoogle Scholar
[4]Berkson, E. and Gillespie, T. A., ‘Spectral decompositions and harmonic analysis on UMD-spaces’, Studia Math. 112 (1994), 13–49.Google Scholar
[5]Blunck, S., ‘Maximal regularity of discrete and continuous time evolution equations’, Studia Math. 146 (2001), 157–163.CrossRefGoogle Scholar
[6]Bourgain, J., ‘Vector-valued singular integrals and the H1-BMO duality’, in: Probability theory and harmonic analysis (ed. Burkholder, D.) (Dekker, New York, 1986) pp. 1–19.Google Scholar
[7]Burkholder, D., ‘A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions’, in: Proc. of Conf. on Harmonic Analysis in Honor of Antoni Zygmund, Chicago 1981 (Wadsworth Publishers, Belmont, CA, 1983) pp. 270–286.Google Scholar
[8]Clément, Ph., de Pagter, B., Sukochev, F. A. and Witvliet, M., ‘Schauder decomposition and multiplier theorems’, Studia Math. 138 (2000), 135–163.Google Scholar
[9]Clément, Ph. and Prüss, J., ‘An operator-valued transference principle and maximal regularity on vector-valued L p-spaces’, in: Evolution equations and their applications in physics and life sciences (eds. Lumer, and Weis, L.), Lecture Notes in Pure and Appl. Math. 215 (Dekker, New York, 2000) pp. 67–87.Google Scholar
[10]Denk, R., Hieber, M. and Prüss, J., ‘R-boundedness, Fourier multipliers and problems of elliptic and parabolic type’, Mem. Amer. Math. Soc. 166 (Amer. Math. Soc., Providence, 2003), p. 788.CrossRefGoogle Scholar
[11]Girardi, M. and Weis, L., ‘Operator-valued Fourier multiplier theorems on Besov spaces’, Math. Nachr. 251 (2003), 34–51.CrossRefGoogle Scholar
[12]Kalton, N. J. and Lancien, G., ‘A solution of the Lp-maximal regularity’, Math. Z. 235 (2000), 559–568.CrossRefGoogle Scholar
[13]Kalton, N. J. and Lancien, G., ‘Lp-maximal regularity on Banach spaces with a Schauder basis’, Arch. Math. (Basel) 78 (2002), 397–408.CrossRefGoogle Scholar
[14]Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces II (Springer, Berlin, 1979).CrossRefGoogle Scholar
[15]Pisier, G., Les inégalités de Khintchine-Kahane d'après C. Borell Séminaire sur la géometrie des espaces de Banach 7 (Ecole Polytechnique Paris, 1977).Google Scholar
[16]Rudin, W., Fourier analysis on groups (Wiley, New York, 1990).CrossRefGoogle Scholar
[17]Štrkalj, Z. and Weis, L., ‘On operator-valued Fourier multiplier theorems’, preprint, 1999.Google Scholar
[18]Weis, L., ‘Operator-valued Fourier multiplier theorems and maximal Lp-regularity’, Math. Ann. 319 (2001), 735–758.CrossRefGoogle Scholar
[19]Zimmermann, F., ‘On vector-valued Fourier multiplier theorems’, Studia Math. 93 (1989), 201–222.CrossRefGoogle Scholar