Toeplitz operators via sesquilinear forms

Book chapter, 2018

The classical theory of Toeplitz operators in spaces of analytic functions deals usually with symbols that are bounded measurable functions on the domain in question. A further extension of the theory was made for symbols being unbounded functions, measures, and compactly supported distributions, all of them subject to some restrictions. In the context of a reproducing kernel Hilbert space we propose a certain common framework, based upon the extensive use of the language of sesquilinear form, for definition of Toeplitz operators for a ‘maximally wide’ class of ‘highly singular’ symbols. Besides covering all previously considered cases, such an approach permits us to introduce a further substantial extension of the class of admissible symbols that generate bounded Toeplitz operators. Although our approach is unified for all reproducing kernel Hilbert spaces, concrete operator consideration are given for Toeplitz operators acting on the standard Fock space, on the standard Bergman space on the unit disk (two leading examples in the classical theory of Toeplitz operators), and on the so-called Herglotz space consisting of the solutions of the Helmholtz equation.