In operator theory, a Toeplitz operator is the compression of a multiplication operator on the circle to the Hardy space.
be the unit circle in the complex plane, with the standard Lebesgue measure, and
be the Hilbert space of complex-valued square-integrable functions.
A bounded measurable complex-valued function
defines a multiplication operator
onto the Hardy space
The Toeplitz operator with symbol
is defined by where " | " means restriction.
A bounded operator on
is Toeplitz if and only if its matrix representation, in the basis
For a proof, see Douglas (1972, 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.
denotes the closed subalgebra of
of analytic functions (functions with vanishing negative Fourier coefficients),
is the closed subalgebra of
is the space (as an algebraic set) of continuous functions on the circle.
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