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Transactions of the American Mathematical Society

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$ p$-absolutely summing operators and the representation of operators on function spaces


Author: John William Rice
Journal: Trans. Amer. Math. Soc. 188 (1974), 53-75
MSC: Primary 47B37
MathSciNet review: 0336429
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Abstract: We introduce a class of p-absolutely summing operators which we call p-extending. We show that for a logmodular function space $ A(K)$, an operator $ T:A(K) \to X$ is p-extending if and only if there exists a probability measure $ \mu $ on K such that T extends to an isometry $ T:{A^p}(K,\mu ) \to X$. We use this result to give necessary and sufficient conditions under which a bounded linear operator is isometrically equivalent to multiplication by z on a space $ {L^p}(K,\mu )$ and certain Hardy spaces $ {H^p}(K,\mu )$.


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  • [1] Joseph Bram, Subnormal operators, Duke Math. J. 22 (1955), 75–94. MR 0068129
  • [2] James E. Brennan, Point evaluations and invariant subspaces, Proceedings of an International Symposium on Operator Theory (Indiana Univ., Bloomington, Ind., 1970), 1971, pp. 879–881. MR 0407640
  • [3] Ion Colojoară and Ciprian Foiaş, Theory of generalized spectral operators, Gordon and Breach, Science Publishers, New York-London-Paris, 1968. Mathematics and its Applications, Vol. 9. MR 0394282
  • [4] J. Diestel, Remarks on the Radon-Nikodym property (to appear).
  • [5] J. Dieudonné, Eléments d'analyse, Tome II, Cahiers Scientifiques, fasc. 31, Gauthier-Villars, Paris, 1968; English transl., Pure and Appl. Math., vols. 10, 11, Academic Press, New York, 1970. MR 38 #4247; 41 #3198.
  • [6] Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR 0117523
  • [7] Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle, Interscience Publishers John Wiley & Sons New York-London, 1963. MR 0188745
  • [8] -, Linear operators. III, Interscience, New York, 1971.
  • [9] C. Foiaş and I. Suciu, On operator representation of logmodular algebras, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968), 505–509 (English, with Loose Russian summary). MR 0238081
  • [10] Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969. MR 0410387
  • [11] A. Grothendieck, Sur les applications linéaires faiblement compactes d’espaces du type 𝐶(𝐾), Canadian J. Math. 5 (1953), 129–173 (French). MR 0058866
  • [12] Paul R. Halmos, A Hilbert space problem book, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0208368
  • [13] Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962. MR 0133008
  • [14] Arnold Lebow, On von Neumann’s theory of spectral sets, J. Math. Anal. Appl. 7 (1963), 64–90. MR 0156220
  • [15] J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in 𝐿_{𝑝}-spaces and their applications, Studia Math. 29 (1968), 275–326. MR 0231188
  • [16] A. Pietsch, Absolut 𝑝-summierende Abbildungen in normierten Räumen, Studia Math. 28 (1966/1967), 333–353 (German). MR 0216328
  • [17] P. G. Spain, On scalar-type spectral operators, Proc. Cambridge Philos. Soc. 69 (1971), 409–410. MR 0290153
  • [18] J. Wermer, Report on subnormal operators, Report of an international conference on operator theory and group representations, Arden House, Harriman, N. Y., 1955, Publ. 387, National Academy of Sciences-National Research Council, Washingtion, D. C., 1955, pp. 1–3. MR 0076316

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0336429-9
Keywords: Banach algebra, functional calculus, p-absolutely summing operator, normal operator, subnormal operator, scalar-type spectral operator, topologically cyclic vector, Hardy space, logmodular, $ {\mathcal{L}_{p,\lambda }}$
Article copyright: © Copyright 1974 American Mathematical Society