View source for Toeplitz operator

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In [[operator theory]], a '''Toeplitz operator''' is the [[dilation (operator theory)|compression]] of a [[multiplication operator]] on the circle to the [[Hardy space]].

==Details==

Let <math>S^1</math> be the unit circle in the complex plane, with the standard Lebesgue measure, and <math>L^2(S^1)</math> be the Hilbert space of complex-valued square-integrable functions. A bounded measurable  complex-valued function <math>g</math> on <math>S^1</math> defines a [[multiplication operator]] <math>M_g</math> on ''<math>L^2(S^1)</math>'' . Let <math>P</math> be the projection from ''<math>L^2(S^1)</math>'' onto the [[Hardy space]] <math>H^2</math>. The ''Toeplitz operator with symbol <math>g</math>'' is defined by

:<math>T_g = P M_g \vert_{H^2},</math>

where " | " means restriction.

A bounded operator on <math>H^2</math> is Toeplitz if and only if its matrix representation, in the [[Basis_(linear_algebra)|basis]] <math>\{z^n, z \in \mathbb{C}, n \geq 0\}</math>, has constant diagonals.
==Theorems==

* Theorem: If <math>g</math> is [[Continuous function|continuous]], then <math>T_g - \lambda</math> is [[Fredholm operator|Fredholm]] if and only if <math>\lambda</math> is not in the set <math>g(S^1)</math>. If it is Fredholm, its index is minus the winding number of the curve traced out by <math>g</math> with respect to the origin.

For a proof, see {{harvtxt|Douglas|1972|loc=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]].

* [[Sheldon Axler|Axler]]-[[Sun-Yung Alice Chang|Chang]]-[[Donald Sarason|Sarason]] Theorem: The operator <math>T_f T_g - T_{fg}</math> is [[Compact operator |compact]] if and only if <math>H^\infty[\bar f] \cap H^\infty [g] \subseteq H^\infty + C^0(S^1)</math>.

Here, <math>H^\infty</math> denotes the closed subalgebra of <math>L^\infty (S^1)</math> of analytic functions (functions with vanishing negative Fourier coefficients), <math>H^\infty [f]</math> is the closed subalgebra of <math>L^\infty (S^1)</math> generated by <math>f </math> and <math> H^\infty</math>, and <math>C^0(S^1)</math> is the space (as an algebraic set) of continuous functions on the circle. See {{harvtxt|S.Axler, S-Y. Chang, D. Sarason |1978}}.

==See also==

* {{annotated link|Toeplitz matrix}}

==References==

{{reflist}}

* {{citation |last= S.Axler, S-Y. Chang, D. Sarason|title=Products of Toeplitz operators|journal=Integral Equations and Operator Theory|volume= 1 |year=1978|issue=3 |pages= 285–309|doi=10.1007/BF01682841 |s2cid=120610368 }}
* {{citation|first1=Albrecht | last1= Böttcher| first2=Sergei M. | last2=Grudsky |title=Toeplitz Matrices, Asymptotic Linear Algebra, and Functional Analysis| url=https://books.google.com/books?id=Dmr0BwAAQBAJ&pg=PA1| year=2000 |publisher=[[Birkhäuser]] |isbn=978-3-0348-8395-5}}.
* {{citation|last1=Böttcher|first1=A.|author1-link=Albrecht Böttcher|last2=Silbermann|first2=B.|year=2006|title=Analysis of Toeplitz Operators|edition=2nd|publisher=Springer-Verlag|series=Springer Monographs in Mathematics|isbn= 978-3-540-32434-8}}.
* {{citation|first=Ronald|last=Douglas|authorlink=Ronald Douglas|title=Banach Algebra techniques in Operator theory|publisher=Academic Press|year=1972}}.
* {{citation|first1=Marvin|last1=Rosenblum|first2=James|last2=Rovnyak|title=Hardy Classes and Operator Theory|year=1985|publisher=Oxford University Press}}. Reprinted by Dover Publications, 1997, {{isbn|978-0-486-69536-5}}.

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[[Category:Operator theory]]
[[Category:Hardy spaces]]
[[Category:Linear operators]]

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