Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Functions of direct integrals of operators


Authors: T. R. Chow and F. Gilfeather
Journal: Proc. Amer. Math. Soc. 29 (1971), 325-330
MSC: Primary 47A60; Secondary 47B15
DOI: https://doi.org/10.1090/S0002-9939-1971-0433239-1
MathSciNet review: 0433239
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper contains two results. The first one is that the unitary dilation of a direct integral of linear contraction operators is the direct integral of unitary dilations. For each linear contraction operator T on a Hilbert space, consider $ f(T)$ as a bounded linear operator. The second result states that if $ T = \smallint \oplus T(s)d\mu (s)$ is decomposable then so is $ f(T)$ and $ f(T) = \smallint \oplus f(T(s))d\mu (s)$.


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

  • [1] T. R. Chow, A spectral theory for the direct integral of operators, Math Ann. 188 (1970), 285-303. MR 0268701 (42:3598)
  • [2] J. Dixmier, Les algèbres d'opérateurs dans l'espace hilbertien, Cahiers Scientifiques, fasc. 25, Gauthier-Villars, Paris, 1957. MR 20 #1234. MR 0094722 (20:1234)
  • [3] N. Dunford, A survey of the theory of spectral operators, Bull. Amer. Math. Soc. 64 (1958), 217-274. MR 21 #3616. MR 0104865 (21:3616)
  • [4] F. Gilfeather, The structure theory of N. Suzuki for non-selfadjoint operators in Hilbert spaces (to appear).
  • [5] B. Sz.-Nagy and C. Foiaş, Analyse harmonique des opérateurs de l'espace de Hilbert, Masson, Paris; Akad. Kiadó, Budapest, 1967. MR 37 #778.
  • [6] M. Schreiber, Unitary dilations of operators, Duke Math. J. 23 (1956), 579-594. MR 18, 748. MR 0083707 (18:748j)
  • [7] -, A functional calculus for general operators in Hilbert spaces, Trans. Amer. Math. Soc. 87 (1958), 108-118. MR 20 #6040. MR 0099601 (20:6040)
  • [8] J. T. Schwartz, $ {W^ \ast }$-algebras, Gordon and Breach, New York, 1967. MR 38 #547. MR 0232221 (38:547)
  • [9] N. Suzuki, The algebraic structure of non-selfadjoint operators, Acta Sci. Math. 27 (1966), 173-184. MR 35 #5971. MR 0215128 (35:5971)
  • [10] J. von Neumann, On rings of operators. Reduction theory, Ann. of Math. (2) 50 (1949), 401-485. MR 10, 548. MR 0029101 (10:548a)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47A60, 47B15

Retrieve articles in all journals with MSC: 47A60, 47B15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0433239-1
Keywords: Direct integrals, primary operators, dilations, strong operator measures and von Neumann algebras
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society