Toeplitz operator

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

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