A dichotomy for linear spaces of Toeplitz operators (Q1270416)

For a Hilbert space \(H\), \(B(H)\) denotes the algebra of all bounded linear operators on \(H\); and the reflexive closure of a subspace \(S\) in \(B(H)\) is defined by: \[ \text{ref}(S)= \{T\in B(H); T(f)\in\overline{S(f)}, \forall f\in H\}. \] \(S\) is said to be reflexive if \(\text{ref}(S)= S\), and transitive if \(\text{ref}(S)= B(H)\). In this paper, the authors proved that: Every intransitive and ultraweakly closed subspace of the space \(F\) of all Toeplitz operators is reflexive. Every proper and ultraweakly closed subspace of \(F\) which contains the algebra of analytic Toeplitz operators is reflexive. This result extends a theorem of D. Sarason: the algebra of analytic Toeplitz operators is reflexive.