Abstract
Let H be a separable, infinite dimensional, complex Hilbert space, and let L(H) denote the algebra of all bounded linear operators on H. The subsets A and A 1 of L(H) (whose definitions are reviewed below), appearing in the theory of dual algebras, were introduced in [4] and studied in several papers during the past three years (cf. [5] for an in-depth development of the theory of dual algebras and a bibliography of pertinent articles). These classes have become important in the study of contraction operators. In particular, it was conjectured in [2] that A = A 1, and if this is true, then an easy corollary is that every contraction T in L(H) such that the spectrum σ(T) of T contains the unit circle T in C has nontrivial invariant subspaces (cf. [2, Conjecture 2.14] and [5, Proposition 4.8]). There are presently several theorems to the effect that if T ε A and has certain additional properties, then T ε A 1 (cf. [5, Chapters VI and VII], [8], and [11]). The conclusion of most of these theorems is the stronger one that T ε A ℵ o (definition reviewed below), but in [11], Sheung, adapting some techniques of [10] to the setting of the functional model of a contraction, gave a nice sufficient condition for membership in A 1 (Theorem 3.1 below) whose conclusion cannot be strengthened. This result of Sheung is also interesting because it seems to be difficult to prove without use of the relatively deep machinery of [10]. In this note we start from Theorem 3.1, and by employing some additional techniques, we arrive at some propositions which we have long thought should be true, and which may be important for the invariant subspace problem for contractions with spectrum equal to T.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Apostol, C.: Ultraweakly closed operator algebras, J. Operator Theory 2(1979), 49–61.
Apostol, C.; Bercovici, H.; Foiaş, C.; Pearcy, C.: Invariant subspaces, dilation theory, and the structure of the predual of a dual algebra, I., J. Fund. Anal. 63(1985), 369–404.
Apostol, C.; Bercovici, H.; Foiaş, C.; Pearcy, C.: Invariant subspaces, dilation theory, and the structure of the predual of a dual algebra, II., Indiana Univ. Math. J. 34(1985), 845–855.
Bercovici, H.; Foiaş, C.; Pearcy, C.: Dilation theory and systems of simultaneous equations in the predual of an operator algebra, I., Michigan Math. J. 30(1983), 335–354.
Bercovici, H.; Foiaş, C.; Pearcy, C.: Dual algebras with applications to invariant subspaces and dilation theory, CBMS Regional Conference Series in Math., No. 56, A.M.S., Providence, 1985.
Brown, S.; Chevreau, B.; Pearcy, C.: Contractions with rich spectrum have invariant subspaces, J. Operator Theory 1(1979), 123–136.
Chevreau, B.; Esterle, J.: Pettis′ lemma and topological properties of dual algebras, Michigan Math. J. 34(1987), 143–146.
Chevreau, B.; Pearcy, C.: On the structure of contraction operators with applications to invariant subspaces, J. Funct. Anal. 67(1986), 360–379.
Kato, T.: Perturbation theory for liner operators, 2nd edition, Springer-Verlag, Berlin, 1966.
Olin, R.; Thompson, J.: Algebras of subnormal operators, J. Funct. Anal. 37(1980), 271–301.
Sheung, J.: On the preduals of certain operator algebras, Ph. D. Thesis, University of Hawaii, 1983.
Westwood, D.: Operators in C 00 with dominating spectrum, J. Funct. Anal. 66(1986), 96–104.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1988 Birkhäuser Verlag Basel
About this chapter
Cite this chapter
Chevreau, B., Pearcy, C. (1988). On Sheung′s Theorem in the Theory of Dual Operator Algebras. In: Arsene, G. (eds) Special Classes of Linear Operators and Other Topics. Operator Theory: Advances and Applications, vol 28. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9164-6_3
Download citation
DOI: https://doi.org/10.1007/978-3-0348-9164-6_3
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-1970-0
Online ISBN: 978-3-0348-9164-6
eBook Packages: Springer Book Archive