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


 * $$T_g = P M_g \vert_{H^2},$$

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. 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.