Toeplitz operator

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 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.

Theorems

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).

Toeplitz operator

Theorems

See also

References

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