Toeplitz operator

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



Let S1 be the circle, with the standard Lebesgue measure, and L2(S1) be the Hilbert space of square-integrable functions. A bounded measurable function g on S1 defines a multiplication operator Mg on L2(S1). Let P be the projection from L2(S1) onto the Hardy space H2. The Toeplitz operator with symbol g is defined by

where " | " means restriction.

A bounded operator on H2 is Toeplitz if and only if its matrix representation, in the basis {zn, n ≥ 0}, has constant diagonals.


For a proof, see Douglas (1972 , p.185). 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, 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).

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