In operator theory

Operator theory

In mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....

, a Toeplitz operator is the compression of a multiplication operator

Multiplication operator

In operator theory, a multiplication operator is a linear operator T defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f...

 on the circle to the Hardy space

Hardy space

In complex analysis, the Hardy spaces Hp are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper...

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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

Multiplication operator

In operator theory, a multiplication operator is a linear operator T defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f...

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


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.