Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

$ 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
DOI: https://doi.org/10.1090/S0002-9947-1974-0336429-9
MathSciNet review: 0336429
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 )$.


References [Enhancements On Off] (What's this?)

  • [1] J. Bram, Subnormal operators, Duke Math. J. 22 (1955), 75-94. MR 16, 835. MR 0068129 (16:835a)
  • [2] J. E. Brennan, Point evaluations and invariant subspaces, Indiana Univ. Math. J. 20 (1971), 879-882. MR 0407640 (53:11412)
  • [3] I. Colojoara and C. Foias, Theory of generalised spectral operators, Gordon and Breach, New York, 1968. MR 0394282 (52:15085)
  • [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] N. Dunford and J. T. Schwartz, Linear operators. I: General theory, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR 22 #8302. MR 0117523 (22:8302)
  • [7] N. Dunford and J. T. Schwartz, Linear operators. II: Spectral theory. Selfadjoint operators in Hilbert space, Interscience, New York, 1963. MR 32 #6181. MR 0188745 (32:6181)
  • [8] -, Linear operators. III, Interscience, New York, 1971.
  • [9] C. Foiaş and I. Suciu, On operator representations of logmodular algebras, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968), 505-509. MR 38 #6357. MR 0238081 (38:6357)
  • [10] T. W. Gamelin, Uniform algebras, Prentice-Hall, Englewood Cliffs, N. J., 1969. MR 0410387 (53:14137)
  • [11] A. Grothendieck, Sur les applications linéaires faiblement compactes, d'espaces du type $ C(K)$, Canad. J. Math. 5 (1953), 129-173. MR 15, 438. MR 0058866 (15:438b)
  • [12] P. Halmos, A Hilbert space problem book, Van Nostrand, Princeton, N. J., 1967. MR 34 #8178. MR 0208368 (34:8178)
  • [13] K. Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Englewood Cliffs, N. J., 1962. MR 24 #A2844. MR 0133008 (24:A2844)
  • [14] A. Lebow, On von Neumann's theory of spectral sets, J. Math. Anal. Appl. 7 (1963), 64-90. MR 27 #6149. MR 0156220 (27:6149)
  • [15] J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in $ {\mathcal{L}_p}$-spaces and their applications, Studia Math. 29 (1968), 275-326. MR 37 #6743. MR 0231188 (37:6743)
  • [16] A. Pietsch, Absolut p-summierende Abbildungen in normierten Raümen, Studia Math. 28 (1966/67), 333-353. MR 35 #7162. MR 0216328 (35:7162)
  • [17] P. G. Spain, On scalar-type spectral operators, Proc. Cambridge Philos. Soc. 69 (1971), 409-410. MR 0290153 (44:7338)
  • [18] J. Wermer, Report on subnormal operators. Report on an International Conference on Operator Theory and Group Representations, Arden House, Harriman, New York, 1955, pp. 1-3; Publ. 387, National Academy of Sciences-National Research Council, Wash., D. C., 1955. MR 17, 880. MR 0076316 (17:880c)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47B37

Retrieve articles in all journals with MSC: 47B37


Additional Information

DOI: https://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

American Mathematical Society