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

Details

defines a multiplication operator on be the projection from onto the Hardy space. The is defined by

where " | " means restriction.

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

Theorems

• is continuous is Fredholm 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 is compact.

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, .