Toeplitz operator explained

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

Details

Let

S ^{1} be the complex unit circle, with the standard Lebesgue measure, and L ^{2(S} ^{1)} be the Hilbert space of square-integrable functions. A bounded measurable function g on S ^{1} defines a multiplication operator M _{g} on L ^{2(S} ^{1)} . Let P be the projection from L ^{2(S} ^{1)} onto the Hardy space H ^{2} . The Toeplitz operator with symbol g is defined byT _{g} = P M _{g} \vert _{H ^{2}} ,

where " | " means restriction.

A bounded operator on

H ^{2} is Toeplitz if and only if its matrix representation, in the basis \{z ^{n,} z \in C , n \geq 0 \} , has constant diagonals.

Theorems

• Theorem: If

g is continuous, then T _{g} - λ is Fredholm if and only if λ is not in the set g(S ^{1)} . If it is Fredholm, its index is minus the winding number of the curve traced out by g 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

T _{f} T _{g} - T _{fg} is compact if and only if H ^{infty[\bar} f] \cap H ^{infty} [g] \subseteq H ^{infty} + C ^{0(S} ^{1)} .

Here,

H ^{infty} denotes the closed subalgebra of L ^{infty} (S ^{1)} of analytic functions (functions with vanishing negative Fourier coefficients), H ^{infty} [f] is the closed subalgebra of L ^{infty} (S ^{1)} generated by f and H ^{infty} , and C ^{0(S} ^{1)} is the space (as an algebraic set) of continuous functions on the circle. See .

References

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• . Reprinted by Dover Publications, 1997, .