Reflexive subspaces of Toeplitz-type operators
by
László Kérchy
University of Szeged, Hungary

In a beautiful paper E. A. Azoff and M. Ptak exhibited a dichotomy property for classical Toeplitz operators. Namely, they showed that every intransitive, weak-* closed subspace of Toeplitz operators is reflexive. Though the set of all Toeplitz operators is transitive, and there are many other transitive subspaces of Toeplitz operators, it turned out that those proper weak-* closed subspaces, which contain the analytic Toeplitz operators, are reflexive. These statements generalize a pioneering result of D. Sarason on the reflexivity of analytic Toeplitz operators.

Our aim was to examine to what extent the theorems and methods of Azoff and Ptak can be extended to the set T(T) of generalized Toeplitz operators, associated with an arbitrary operator T having a regular norm-sequence {||Tⁿ||}_n=1^\infty. An operator A belongs to T(T), and is called T-Toeplitz, if T^*AT=r(T)²A. An appropriate symbolic calculus can be given for T(T) in terms of operators in the commutant of a unitary operator U_T. It turns out that a quasianalytic property is responsible for the transitivity of T(T). If the vectors g, h have an analytic property with respect to T, then the weak-* closed subspace T_{g, h}(T)={A in T(T): <Ag, h> = 0} is elementary. Furthermore, if g and h have an inner-outer-type factorization with respect to T, then the space T_{g, h}(T) is reflexive. Since the elementary and reflexive subspaces are hereditarily reflexive, we obtain generalizations of the results due to Azoff and Ptak.

Date received: June 7, 2002