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## Summary

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

## Details

Let S1 be the circle, with the standard Lebesgue measure, and L2(S1) be the Hilbert space of square-integrable functions. A bounded measurable function g on S1 defines a multiplication operator Mg on L2(S1). Let P be the projection from L2(S1) onto the Hardy space H2. The Toeplitz operator with symbol g is defined by

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

where " | " means restriction.

A bounded operator on H2 is Toeplitz if and only if its matrix representation, in the basis {zn, n ≥ 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 $H^{\infty }[{\bar {f}}]\cap H^{\infty }[g]\subseteq H^{\infty }+C$ .

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 $C$  is the continuous functions on the circle. See S.Axler, S-Y. Chang, D. Sarason (1978).