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

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

where " | " means restriction.

A bounded operator on ''H''<sup>2</sup> is Toeplitz if and only if its matrix representation, in the basis {''z<sup>n</sup>'', ''n'' ≥ 0}, has constant diagonals.

== References ==
*{{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|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]]

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