# 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(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

${\displaystyle T_{g}=PM_{g}\vert _{H^{2}},}$

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.