Bernard Chevreau

Dilation Theory and Systems of Simultaneous Equations in the Predual of an Operator Algebra. II

1. This note is a continuation of our earlier paper [-3], in which we developed a dilation theory for a certain class of contraction operators acting on a separable, infinite dimensional, complex Hilbert space ~ . The notation and terminology in what follows is taken from [3]. For the convenience of the reader we recall a few pertinent definitions. The algebra of bounded linear operators on ~ is denoted by Y ( ~ ) . If T~Se(2C), the ultraweakly closed algebra generated by T and l~e is denoted by dr; we recall that d r can be identified with the dual space of the quotient space Q r = ( z c ) / • where (zc) denotes the ideal of trace-class operators in 5~(24 ~) and • is the preannihilator of d r in (z c), under the pairing

Operators that admit a moment sequence, II

We consider the question of which operators on Hilbert space admit a moment sequence, and show, in particular, that ifT is any operator inL(itH) that can be written asT=N+K, whereN is normal andK is compact, andM⊂itH is any invariant subspace forT, then the restrictionT/itM admits such a sequence.

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

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.

Recent Results on Reflexivity of Operators and Algebras of Operators

Let H be a complex Hilbert space and let L(H) denote the algebra of all bounded linear operators on H. For any subset S of L (H) let, as usual, Lat (S) be the set of (closed) subspaces M of H invariant under any element of S (i.e. TM⊂M for any T e S) and A1gLat(S) the subalgebra of L(H) consisting of those operators T such that Lat(T)⊃ ⊃Lat(S). Of course Alg Lat(S) is closed in the weak operator topology of L(H) (WOT for short). A (necessarily WOT-closed) subalgebra A of L(H) is reflexive if A = AlgLat(A). An operator T is reflexive if WT (the unital WOT-closed algebra generated by T) is reflexive. Thus roughly speaking an operator is reflexive if its lattice of invariant subspaces is rich enough so as to determine the WOT-closed subalgebra it generates. This is indirectly confirmed by the fact that a unicellular operator (i.e. an operator T such that Lat(T) is linearly ordered) cannot be reflexive.