In operator theory, a Toeplitz operator is the compression of a multiplication operator on the circle to the Hardy space.
Details[edit]
Let be the complex unit circle, with the standard Lebesgue measure, and be the Hilbert space of square-integrable functions. A bounded measurable 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[edit]
- 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.
![{\displaystyle H^{\infty }[{\bar {f}}]\cap H^{\infty }[g]\subseteq H^{\infty }+C^{0}(S^{1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c516b6496f1bf8b51669c2b6fbdf7595e38aba25)
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).
![{\displaystyle H^{\infty }[f]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e6ad23f09ddac57993da2d1ab6fcce2d5f01300)
See also[edit]
- Toeplitz matrix – Matrix with shifting rows
References[edit]
- S.Axler, S-Y. Chang, D. Sarason (1978), "Products of Toeplitz operators", Integral Equations and Operator Theory, 1 (3): 285–309, doi:10.1007/BF01682841, S2CID 120610368
{{citation}}: CS1 maint: multiple names: authors list (link) - Böttcher, Albrecht; Grudsky, Sergei M. (2000), Toeplitz Matrices, Asymptotic Linear Algebra, and Functional Analysis, Birkhäuser, ISBN 978-3-0348-8395-5.
- Böttcher, A.; Silbermann, B. (2006), Analysis of Toeplitz Operators, Springer Monographs in Mathematics (2nd ed.), Springer-Verlag, ISBN 978-3-540-32434-8.
- Douglas, Ronald (1972), Banach Algebra techniques in Operator theory, Academic Press.
- Rosenblum, Marvin; Rovnyak, James (1985), Hardy Classes and Operator Theory, Oxford University Press. Reprinted by Dover Publications, 1997, ISBN 978-0-486-69536-5.