Toeplitz operator

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

Details

Let be the unit circle in the complex plane, with the standard Lebesgue measure, and be the Hilbert space of complex-valued square-integrable functions. A bounded measurable complex-valued function on defines a multiplication operator on ' . Let be the projection from ' onto the Hardy space . The Toeplitz operator with symbol is defined by

where " | " means restriction.

A bounded operator on is Toeplitz if and only if its matrix representation, in the basis , has constant diagonals.

Theorems

• Theorem: If is continuous, then is Fredholm if and only if is not in the set . If it is Fredholm, its index is minus the winding number of the curve traced out by with respect to the origin.

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.

• Axler-Chang-Sarason Theorem: The operator is compact if and only if .

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 .

See also

References

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• . Reprinted by Dover Publications, 1997, .