  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.
