On Sheung′s Theorem in the Theory of Dual Operator Algebras

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 e A and has certain additional properties, then T e A 1 (cf. [5, Chapters VI and VII], [8], and [11]). The conclusion of most of these theorems is the stronger one that T e 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.