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

$T_g=P{M}_g{|}_{H^2},$

where " | " means restriction.

A bounded operator on $H^2$ is Toeplitz if and only if its matrix representation, in the basis $\{z^n,z\in \mathbb{ℂ},n\ge 0\}$, 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). 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 $H^{\mathrm{\infty }}[\overline{f}]\cap H^{\mathrm{\infty }}[g]\subseteq H^{\mathrm{\infty }}+C^0(S^1)$.

Here, $H^{\mathrm{\infty }}$ denotes the closed subalgebra of $L^{\mathrm{\infty }}(S^1)$ of analytic functions (functions with vanishing negative Fourier coefficients), $H^{\mathrm{\infty }}[f]$ is the closed subalgebra of $L^{\mathrm{\infty }}(S^1)$ generated by $f$ and $H^{\mathrm{\infty }}$, 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).

See also

• Toeplitz matrix – Matrix with equal values along diagonals

References

• S.Axler, S-Y. Chang, D. Sarason (1978), "Products of Toeplitz operators", Integral Equations and Operator Theory 1 (3): 285–309, doi:10.1007/BF01682841 
• Böttcher, Albrecht; Grudsky, Sergei M. (2000), Toeplitz Matrices, Asymptotic Linear Algebra, and Functional Analysis, Birkhäuser, ISBN 978-3-0348-8395-5, https://books.google.com/books?id=Dmr0BwAAQBAJ&pg=PA1 .
• Böttcher, A.; Silbermann, B. (2006), Analysis of Toeplitz Operators, Springer Monographs in Mathematics (2nd ed.), Springer-Verlag, ISBN 978-3-540-32434-8 .
• Douglas, Ronald (1972), Banach Algebra techniques in Operator theory, Academic Press .
• Rosenblum, Marvin; Rovnyak, James (1985), Hardy Classes and Operator Theory, Oxford University Press . Reprinted by Dover Publications, 1997, ISBN 978-0-486-69536-5.