Toeplitz operators generalise matrices which are constant on diagonals. There is a well-developed theory of these operators, particularly when acting on the Hardy space H2(T)H2(T) which might be thought of as consisting of functions on the unit circle, or else analytic functions on the unit disk. These operators acting on H2(T)H2(T) have an interesting connection with the theory of Fourier series, which motivated their study. A less well-known space is the Segal-Bargmann space which was motivated by work in Quantum Mechanics. This Hilbert space consists of analytic functions on CnCn that do not grow too rapidly.
"We investigate Toeplitz and Hankel operators acting on the Segal-Bargmann space, and determine necessary and sufficient conditions for these operators to be compact. The sub-algebra of L∞(Cn)L∞(Cn) consisting of `bounded continuous eventually slowly varying' functions can be used to determine which Toeplitz operators are Fredholm. These functions establish an index theorem for their corresponding Toeplitz operators. This theorem is analogous to the index theorem for Toeplitz operators acting on the Hardy space H2(T)H2(T), which characterises index in terms of winding number.