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 ${z^{n}, z \in ℂ, n \geq 0}$ , has constant diagonals.

Theorems

Theorem: If $g$ is continuous, then $T_{g} - λ$ is Fredholm if and only if $λ$ 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 $T_{f} T_{g} - T_{f g}$ 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).

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, S2CID 120610368{{citation}}: CS1 maint: multiple names: authors list (link)
Böttcher, Albrecht; Grudsky, Sergei M. (2000), Toeplitz Matrices, Asymptotic Linear Algebra, and Functional Analysis, Birkhäuser, ISBN 978-3-0348-8395-5.
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.
V. V. Peller, Hankel Operators and their Applications, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003

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Toeplitz operator

Details

Theorems

See also

References