# .

# 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 circle, with the standard Lebesgue measure, and *L*^{2}(*S*^{1}) be the Hilbert space of square-integrable functions. A bounded measurable 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

\( T_g = P M_g \vert_{H^2}, \)

where " | " means restriction.

A bounded operator on *H*^{2} is Toeplitz if and only if its matrix representation, in the basis {*z ^{n}*,

*n*≥ 0}, has constant diagonals.

References

Böttcher, A.; Silbermann, B. (2006), Analysis of Toeplitz Operators, Springer Monographs in Mathematics (2nd ed.), Springer-Verlag, ISBN 9783540324348.

Rosenblum, Marvin; Rovnyak, James (1985), Hardy Classes and Operator Theory, Oxford University Press. Reprinted by Dover Publications, 1997, ISBN 9780486695365.

cylindrical symmetry without a symmetry plane perpendicular to the axis, this applies for example often for a bottle

cylindrical symmetry with a symmetry plane perpendicular to the axis

Retrieved from "http://en.wikipedia.org/"

All text is available under the terms of the GNU Free Documentation License