Toeplitz operator


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

Details

Let S1 be the circle, with the standard Lebesgue measure, and L2 be the Hilbert space of square-integrable functions. A bounded measurable function g on S1 defines a multiplication operator Mg on L2. Let P be the projection from L2 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, 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, denotes the closed subalgebra of of analytic functions
,
is the closed subalgebra of generated by and
, and is the continuous functions on the circle.
See