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In the paper we develop the Gelfand theory for a new class of commutative Banach algebras generated by Toeplitz operators both in the Bergman and the Hardy space cases. We introduce the so-called generalized pseudo-homogeneous symbols, the functions of the form, where, thus, and. This class of symbols is a result of step-by-step generalization of previously considered (and studied) symbols. For the commutative algebras under study we describe the compact sets of their maximal ideals, Gelfand transform, and give a criterion on whether our algebra is semi-simple or has a radical. In the last case, we give a complete characterization of the radical.

Toeplitz operators on the Hardy space with generalized pseudo-homogeneous symbols