Toeplitz operator

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In operator theory, a Toeplitz operator is the compression of a multiplication operator on the circle to the Hardy space.


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 Mg 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

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

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.


  • Böttcher, A.; Silbermann, B. (2006), Analysis of Toeplitz Operators, Springer Monographs in Mathematics (2nd ed.), Springer-Verlag, ISBN 978-3-540-32434-8 .
  • Rosenblum, Marvin; Rovnyak, James (1985), Hardy Classes and Operator Theory, Oxford University Press . Reprinted by Dover Publications, 1997, ISBN 978-0-486-69536-5.