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

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.

• 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.

• 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 S.Axler, S-Y. Chang, D. Sarason (1978).

• Toeplitz matrix – Matrix with shifting rows

• S.Axler, S-Y. Chang, D. Sarason (1978), "Products of Toeplitz operators", Integral Equations and Operator Theory, 1 (3): 285–309, doi:10.1007/BF01682841, S2CID 120610368{{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.