Encyclopedia

Toeplitz operator

Wikipedia contributorsAdapted for Fathomly · View authors

1 min read
Source updated August 1, 2026

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

01Details

Let $S^{1}$ be the unit circle in the complex plane, with the standard Lebesgue measure, and $L^{2}(S^{1})$ be the Hilbert space of complex-valued square-integrable functions. A bounded measurable complex-valued 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 by

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

where " | " means restriction.

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

02Theorems

Theorem: If $g$ is continuous, then $T_{g}-\lambda$ is Fredholm if and only if $\lambda$ 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 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 $T_{f}T_{g}-T_{fg}$ 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 S.Axler, S-Y. Chang, D. Sarason (1978).

Watch videos about Toeplitz operatorExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Toeplitz operator”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.

Toeplitz operator

01Details

02Theorems

Sources and credits

Related topics

Toeplitz matrix