Toeplitz operators on bounded symmetric domains

Abstract

In this paper Jordan algebraic methods are applied to study Toeplitz operators on the Hardy space H 2 ( S ) {H^2}(S) associated with the Shilov boundary S S of a bounded symmetric domain D D in C n {{\mathbf {C}}^n} of arbitrary rank. The Jordan triple system Z ≈ C n Z \approx {{\mathbf {C}}^n} associated with D D is used to determine the relationship between Toeplitz operators and differential operators. Further, it is shown that each Jordan triple idempotent e ∈ Z e \in Z induces an irreducible representation (" e e -symbol") of the C ∗ {C^{\ast } } -algebra T \mathcal {T} generated by all Toeplitz operators T f {T_f} with continuous symbol function f f .