Let be the unit circle in the complex plane, with the standard Lebesgue measure, and be the Hilbert space of complex-valued square-integrable functions. A bounded measurable complex-valued function on defines a multiplication operator on . Let be the projection from onto the Hardy space. The Toeplitz operator with symbol is defined by
where " | " means restriction.
A bounded operator on is Toeplitz if and only if its matrix representation, in the basis, has constant diagonals.
Theorems
Theorem: If is continuous, then is Fredholm if and only if is not in the set . If it is Fredholm, its index is minus the winding number of the curve traced out by with respect to the origin.
Here, denotes the closed subalgebra of of analytic functions (functions with vanishing negative Fourier coefficients), is the closed subalgebra of generated by and , and is the space (as an algebraic set) of continuous functions on the circle. See S.Axler, S-Y. Chang, D. Sarason (1978).
Böttcher, A.; Silbermann, B. (2006), Analysis of Toeplitz Operators, Springer Monographs in Mathematics (2nd ed.), Springer-Verlag, ISBN978-3-540-32434-8.
Douglas, Ronald (1972), Banach Algebra techniques in Operator theory, Academic Press.
Rosenblum, Marvin; Rovnyak, James (1985), Hardy Classes and Operator Theory, Oxford University Press. Reprinted by Dover Publications, 1997, ISBN978-0-486-69536-5.