In operator theory, a Toeplitz operator is the compression of a multiplication operator on the circle to the Hardy space.

Details

Let S^1 be the complex unit circle, with the standard Lebesgue measure, and L^2(S^1) be the Hilbert space of square-integrable functions. A bounded measurable function g on S^1 defines a multiplication operator M_g on L^2(S^1). Let P be the projection from L^2(S^1) onto the Hardy space H^2. The Toeplitz operator with symbol g is defined by where " | " means restriction. A bounded operator on H^2 is Toeplitz if and only if its matrix representation, in the basis, has constant diagonals.

Theorems

For a proof, see. He attributes the theorem to Mark Krein, Harold Widom, and Allen Devinatz. This can be thought of as an important special case of the Atiyah-Singer index theorem. Here, H^\infty denotes the closed subalgebra of of analytic functions (functions with vanishing negative Fourier coefficients), is the closed subalgebra of generated by f and H^\infty, and C^0(S^1) is the space (as an algebraic set) of continuous functions on the circle. See.