A Dichotomy for Linear Spaces of Toeplitz Operators

LetSbe a linear manifold of bounded Hilbert space operators. An operatorAbelongs to thereflexive closureofSifAfbelongs to the closure ofSffor each vectorfin the underlying Hilbert space. Two extreme possibilities are (1)Sisreflexivein the sense that refS=S, and (2)Sistransitivein the sense that refSincludes all bounded operators on the underlying space. We show that every linear space B of Toeplitz operators which is closed in the ultraweak operator topology is either transitive or reflexive. No intermediate behavior is possible. The full space of all Toeplitz operators is transitive, but if B is properly contained in this space and contains allanalyticToeplitz operators, then B must be reflexive. In particular, the space of Toeplitz operators whose matrices have zeros on a fixed superdiagonal is reflexive.