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 M_g 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

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

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.

References

• Böttcher, A. and Silbermann, B. (2006). Analysis of Toeplitz Operators, second edn, Springer.
• Marvin Rosenblum and James Rovnyak, Hardy Classes and Operator Theory, (1985) Oxford University Press.