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.
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Details
Summarize
Perspective
Let 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 on defines a multiplication operator on . Let be the projection from 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 , has constant diagonals.
Theorems
Summarize
Perspective
- Theorem: If is continuous, then is Fredholm if and only if is not in the set . If it is Fredholm, its index is minus the winding number of the curve traced out by with respect to the origin.
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.
Here, denotes the closed subalgebra of of analytic functions (functions with vanishing negative Fourier coefficients), is the closed subalgebra of generated by and , and is the space (as an algebraic set) of continuous functions on the circle. See S.Axler, S-Y. Chang, D. Sarason (1978).
See also
- Toeplitz matrix – Matrix with shifting rows
References
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