Toeplitz Operators Defined by Sesquilinear Forms: Bergman Space Case

Abstract

Abstract. The definition of Toeplitz operators in the Bergman space A 2 (D) of square integrable analytic functions in the unit disk in the complex plane is extended in such way that it covers many cases where the traditional definition does not work. This includes, in particular, highly singular symbols such as measures, distributions, and certain hyper-functions.

Toeplitz Operators in the Herglotz Space

“…This approach does not require any enveloping Hilbert space, it uses the sesquilinear form and the reproducing kernel only. It has been proposed by the authors in [25], having been applied to Toeplitz operators in the Fock space over the complex plane C and developed further in [26] for the Bergman space case on the disk in C. In addition to eliminating the need of an enveloping space, this approach enabled us to consider Toeplitz operators with highly singular symbols, involving measures, distributions and even certain hyper-functions. As usual, a bounded sesquilinear form F(u, v) in H is linear in u, anti-linear in v, and satisfies |F(u, v)| ≤ C u v for all u, v ∈ H. As explained in [25], having such a bounded sesquilinear form F, the Toeplitz operator T F , with form-symbol F, in H is defined by…”

“…A general unified approach to defining Toeplitz operators, based upon the reproducing kernel subspaces, was developed in [25], [26]. The role of B can be played by any reproducing kernel space while the symbol is realized by a bounded sesquilinear form F(u, v), u, v ∈ B.…”

“…Viability of such approach was demonstrated in [25], [26] for Toeplitz operators acting on the classical Bergman space on the unit disk and the Fock space. As was shown there, definition (1.2) covers (apart of the standard classical case: bounded measurable function symbols) a large variety of singular symbols, such as unbounded functions, measures, distributions, and even hyperfunctions in a certain class.…”