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

Let *S*^{1} be the 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

where " | " means restriction.

A bounded operator on *H*^{2} is Toeplitz if and only if its matrix representation, in the basis {*z ^{n}*,

- 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 Douglas (1972, p.185). 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.

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 continuous functions on the circle. See S.Axler, S-Y. Chang, D. Sarason (1978).

- S.Axler, S-Y. Chang, D. Sarason (1978), "Products of Toeplitz operators",
*Integral Equations and Operator Theory*,**1**: 285–309`{{citation}}`

: CS1 maint: multiple names: authors list (link) - Böttcher, Albrecht; Grudsky, Sergei M. (2000),
*Toeplitz Matrices, Asymptotic Linear Algebra, and Functional Analysis*, Birkhäuser, ISBN 978-3-0348-8395-5. - Böttcher, A.; Silbermann, B. (2006),
*Analysis of Toeplitz Operators*, Springer Monographs in Mathematics (2nd ed.), Springer-Verlag, ISBN 978-3-540-32434-8. - Douglas, Ronald (1972),
*Banach Algebra techniques in Operator theory*, Academic Press. - Rosenblum, Marvin; Rovnyak, James (1985),
*Hardy Classes and Operator Theory*, Oxford University Press. Reprinted by Dover Publications, 1997, ISBN 978-0-486-69536-5.