Toeplitz operator

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 M_g 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

[math]\displaystyle{ T_g = P M_g \vert_{H^2}, }[/math]

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 [math]\displaystyle{ g }[/math] is continuous, then [math]\displaystyle{ T_g - \lambda }[/math] is Fredholm if and only if [math]\displaystyle{ \lambda }[/math] is not in the set [math]\displaystyle{ g(S^1) }[/math]. If it is Fredholm, its index is minus the winding number of the curve traced out by [math]\displaystyle{ g }[/math] 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 [math]\displaystyle{ T_f T_g - T_{fg} }[/math] is compact if and only if [math]\displaystyle{ H^\infty[\bar f] \cap H^\infty [g] \subseteq H^\infty + C^0(S^1) }[/math].

Here, [math]\displaystyle{ H^\infty }[/math] denotes the closed subalgebra of [math]\displaystyle{ L^\infty (S^1) }[/math] of analytic functions (functions with vanishing negative Fourier coefficients), [math]\displaystyle{ H^\infty [f] }[/math] is the closed subalgebra of [math]\displaystyle{ L^\infty (S^1) }[/math] generated by [math]\displaystyle{ f }[/math] and [math]\displaystyle{ H^\infty }[/math], and [math]\displaystyle{ C^0(S^1) }[/math] 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.