James E. Thomson

Algebras of subnormal operators

The paper deals with the following: (I) If S is a subnormal operator on H, then Ol(S) = W(S) = Alg Lat S. (II) If L ∈ (Ol(S), σ-wot)∗, then there exist vectors a and b in H such that L(T) = 〈T a, b〉 for every T in Ol. (III) In addition to I the map i(T) = T is a homeomorphism from (Ol, σ-wot) onto (W(S), wot). (IV) If S is not a reductive normal operator, then there exists a cyclic invariant subspace for S that has an open set of bounded point evaluations. (This open set can be constructed to be as large as possible.)

Cellular-indecomposable subnormal operators

A bounded operator T is cellular-indecomposable if LnM≠{0} whenever L and M are any two nonzero invariant subspaces for T. We show that any such subnormal operator has a cyclic normal extension and is unitarily equivalent modulo the compact operators to an analytic Toeplitz operator whose symbol is a weak-star generator of H∞.

A local lifting theorem for subnormal operators

Criteria for the existence of lifts of operators intertwining subnormal operators are established. The main result of the paper reduces lifting questions for general subnormal operators to questions about lifts of cyclic subnormal operators. It is shown that in general the existence of local lifts (i.e. those coming from cyclic parts) for a pair of subnormal operators does not imply the existence of a global lift. However this is the case when minimal normal extensions of subnormal operators in question are star-cyclic.

A note on density for the core of unbounded Bergman operators

In this paper, we identify a large collection of open subsets of the complex plane for which the core of corresponding unbounded Bergman operators is densely defined. This result gives the necessary background to investigate the concept of invariant subspaces, index, and cyclicity in the unbounded case.