Toeplitz Operator

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

Details

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

:$T_g = P M_g \vert_,$

where " , " means restriction.

A bounded operator on $H^2$ is Toeplitz if and only if its matrix representation, in the basis $\$, 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 . 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 $T_f T_g - T_$ is compact if and only if H^\infty bar f\cap H^\infty \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 /math> 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 .

See also

*

References

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* . Reprinted by Dover Publications, 1997, .

Operator theory
Hardy spaces
Linear operators

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