Toeplitz operators on the symmetrized bidisc

Abstract: The symmetrized bidisc has been a rich field of holomorphic function theory and operator theory. A certain well-known reproducing kernel Hilbert space of holomorphic functions on the symmetrized bidisc resembles the Hardy space of the unit disc in several aspects. This space is known as the Hardy space of the symmetrized bidisc. We introduce the study of those operators on the Hardy space of the symmetrized bidisc that are analogous to Toeplitz operators on the Hardy space of the unit disc. More explicitly, we first study multiplication operators on a bigger space (an $L^2$-space) and then study compressions of these multiplication operators to the Hardy space of the symmetrized bidisc and prove the following major results: (1) Theorem I analyzes the Hardy space of the symmetrized bidisc, not just as a Hilbert space, but as a Hilbert module over the polynomial ring and finds three isomorphic copies of it as $\mathbb D^2$-contractive Hilbert modules. (2) Theorem II provides an algebraic, Brown and Halmos type, characterization of Toeplitz operators. (3)Theorem III gives several characterizations of an analytic Toeplitz operator. (4)Theorem IV characterizes asymptotic Toeplitz operators. (5)Theorem V is a commutant lifting theorem. (6)Theorem VI yields an algebraic characterization of dual Toeplitz operators. Every section from Section 1 to Section 6 contains a theorem each, the main result of that section.