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

${\displaystyle T_{g}=PM_{g}\vert _{H^{2}},}$ 

where " | " means restriction.

A bounded operator on ${\displaystyle H^{2}}$  is Toeplitz if and only if its matrix representation, in the basis ${\displaystyle \{z^{n},z\in \mathbb {C} ,n\geq 0\}}$ , has constant diagonals.

• Theorem: If ${\displaystyle g}$  is continuous, then ${\displaystyle T_{g}-\lambda }$  is Fredholm if and only if ${\displaystyle \lambda }$  is not in the set ${\displaystyle g(S^{1})}$ . If it is Fredholm, its index is minus the winding number of the curve traced out by ${\displaystyle g}$  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 ${\displaystyle T_{f}T_{g}-T_{fg}}$  is compact if and only if ${\displaystyle H^{\infty }[{\bar {f}}]\cap H^{\infty }[g]\subseteq H^{\infty }+C^{0}(S^{1})}$ .

Here, ${\displaystyle H^{\infty }}$  denotes the closed subalgebra of ${\displaystyle L^{\infty }(S^{1})}$  of analytic functions (functions with vanishing negative Fourier coefficients), ${\displaystyle H^{\infty }[f]}$  is the closed subalgebra of ${\displaystyle L^{\infty }(S^{1})}$  generated by ${\displaystyle f}$  and ${\displaystyle H^{\infty }}$ , and ${\displaystyle C^{0}(S^{1})}$  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.