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1991-1992 | 101 | 2 | 193-214
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Decomposable multipliers and applications to harmonic analysis

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EN
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EN
For a multiplier on a semisimple commutative Banach algebra, the decomposability in the sense of Foiaş will be related to certain continuity properties and growth conditions of its Gelfand transform on the spectrum of the multiplier algebra. If the multiplier algebra is regular, then all multipliers will be seen to be decomposable. In general, an important tool will be the hull-kernel topology on the spectrum of the typically nonregular multiplier algebra. Our investigation involves various closed subalgebras of the multiplier algebra and includes perturbation results of Wiener-Pitt type for the invertibility of multipliers. Under suitable topological assumptions on the spectrum of the given Banach algebra, we shall characterize decomposable multipliers, Riesz multipliers, and multipliers with natural or countable spectrum. Most of these results are new even in the case of convolution operators given by measures on a locally compact abelian group. We shall obtain various classes of measures for which the corresponding convolution operators are decomposable both on the measure algebra and on the group algebra. Moreover, the spectral properties of a convolution operator will be related to the behavior of the Fourier-Stieltjes transform of the underlying measure on the dual group and on the spectrum of the measure algebra. Finally, it will be shown that the decomposability of convolution operators behaves nicely with respect to absolute continuity and singularity of measures.
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Twórcy
  • Mathematical Institute, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark
  • Department of Mathematics and Statistics, Mississippi State University, P. O. Drawer MA, Mississippi State, Mississippi 39762, U.S.A.
Bibliografia
  • [1] P. Aiena, A characterization of Riesz operators, Math. Z. 196 (1987), 491-496.
  • [2] P. Aiena, Riesz multipliers on L₁(G), Arch. Math. (Basel) 50 (1988), 459-462.
  • [3] P. Aiena, Riesz multipliers on commutative semisimple Banach algebras, ibid. 54 (1990), 293-303.
  • [4] C. A. Akemann, Some mapping properties of the group algebras of a compact group, Pacific J. Math. 22 (1967), 1-8.
  • [5] E. Albrecht, Decomposable systems of operators in harmonic analysis, in: Toeplitz Centennial, Birkhäuser, Basel 1982, 19-35.
  • [6] C. Apostol, Decomposable multiplication operators, Rev. Roumaine Math. Pures Appl. 17 (1972), 323-333.
  • [7] A. Beurling, Sur les intégrales de Fourier absolument convergentes et leurs application à une transformation fonctionnelle, in: Proc. Neuv. Congr. Math. Scand. Helsingfors 1938, Mercator, Helsingfors 1939, 345-366.
  • [8] F. T. Birtel, Banach algebras of multipliers, Duke Math. J. 28 (1961), 203-211.
  • [9] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, New York 1973.
  • [10] I. Colojoară and C. Foiaş, Theory of Generalized Spectral Operators, Gordon and Breach, New York 1968.
  • [11] J. Eschmeier, Operator decomposability and weakly continuous representations of locally compact abelian groups, J. Operator Theory 7 (1982), 201-208.
  • [12] C. C. Graham and O. C. McGehee, Essays in Commutative Harmonic Analysis, Springer, New York 1979.
  • [13] S. Hartman, Beitrag zur Theorie des Maßringes mit Faltung, Studia Math. 18 (1959), 67-79.
  • [14] H. Heuser, Functional Analysis, Wiley, New York 1982.
  • [15] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. II, Springer, New York 1970.
  • [16] E. Hewitt and H. S. Zuckerman, Singular measures with absolutely continuous convolution squares, Proc. Cambridge Philos. Soc. 62 (1966), 399-420; Corrigendum, ibid. 63 (1967), 367-368.
  • [17] R. Larsen, An Introduction to the Theory of Multipliers, Springer, New York 1971.
  • [18] K. B. Laursen and M. M. Neumann, Decomposable operators and automatic continuity, J. Operator Theory 15 (1986), 33-51.
  • [19] M. M. Neumann, Commutative Banach algebras and decomposable operators, preprint, Mississippi State University, 1990, 17 pp. (submittted).
  • [20] M. M. Neumann, Banach algebras, decomposable convolution operators, and a spectral mapping property, in: Proceedings of the Conference on Function Spaces at Southern Illinois University at Edwardsville, Marcel Dekker, New York 1991, 307-323.
  • [21] A. Pełczyński and Z. Semadeni, Spaces of continuous functions (III), Studia Math. 18 (1959), 211-222.
  • [22] C. E. Rickart, General Theory of Banach Algebras, Van Nostrand, Princeton, N.J., 1960. Reprinted by R. E. Krieger Publ. Company, Huntington, N.Y., 1974.
  • [23] W. Rudin, Continuous functions on compact spaces without perfect subsets, Proc. Amer. Math. Soc. 8 (1957), 39-42.
  • [24] W. Rudin, Fourier Analysis on Groups, Interscience Publ., New York 1962.
  • [25] F.-H. Vasilescu, Analytic Functional Calculus and Spectral Decompositions, Editura Academiei and D. Reidel, Bucureşti and Dordrecht 1982.
  • [26] N. Wiener and H. R. Pitt, On absolutely convergent Fourier-Stieltjes transforms, Duke Math. J. 4 (1938), 420-436.
  • [27] M. Zafran, On the spectra of multipliers, Pacific J. Math. 47 (1973), 609-626.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-smv101i2p193bwm
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