Toeplitz operators defined by sesquilinear forms: Fock space case

Toeplitz Operators Defined by Sesquilinear Forms: Bergman Space Case

“…We recall here the general approach [15] to the extension of the notion of Bergman-Toeplitz (just Toeplitz in what follows) operators for a wide class of singular symbols, given in the context of reproducing kernel Hilbert spaces.…”

“…Concrete operator considerations of thus defined Toeplitz operators were elaborated in [15] for operators acting in the classical Fock (or Segal-Bargmann) space. In the present paper we specify our approach to the operators acting in the Bergman space A 2 (D) over the unit disk D in C. This Bergman space was being studied for a long time, from different points of view, with numerous applications; we refer, in particular, to the books [7], [22].…”

“…These latter Toeplitz operators are called the Bergman-Toeplitz ones. The present paper is a continuation of [15], and is devoted to the development of the theory of the Bergman-Toeplitz operators acting on the Bergman space of analytic functions on the unit disk D.…”

“…A far going generalization of the traditional approach to Toeplitz operators has been developed in [15], where the notion of the sesquilinear form symbol was introduced. The concrete study was concentrated there around operators in the Fock space, the space of entire analytic function in C, square integrable against the Gaussian measure.…”

“…In the present paper we follow the pattern of [15] and consider Toeplitz operator in another leading case, the Bergman space of analytic function in the unit disk, square integrable with respect to the Lebesgue measure. Having defined the general sesquilinear forms as symbols, we present a number of particular classes of symbols, including the ones absent in the Fock space case.…”