Toeplitz operator

Let $S^1$ be the unit circle in the complex plane, with the standard Lebesgue measure, and $L^2(S^1)$ be the Hilbert space of complex-valued square-integrable functions. A bounded measurable complex-valued 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

:$T\_g\; =\; P\; M\_g\; \backslash vert\_\{H^2\},$

where " | " means restriction.

A bounded operator on $H^2$ is Toeplitz if and only if its matrix representation, in the basis $\backslash \{z^n,\; z\; \backslash in\; \backslash mathbb\{C\},\; n\; \backslash geq\; 0\backslash \}$, has constant diagonals.

• Theorem: If $g$ is continuous, then $T\_g\; -\; \backslash lambda$ is Fredholm if and only if $\backslash lambda$ is not in the set $g(S^1)$. If it is Fredholm, its index is minus the winding number of the curve traced out by $g$ with respect to the origin.

For a proof, see {{harvtxt|Douglas|1972|loc=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.

• Axler-Chang-Sarason Theorem: The operator $T\_f\; T\_g\; -\; T\_\{fg\}$ is compact if and only if $H^\backslash infty[\backslash bar\; f]\; \backslash cap\; H^\backslash infty\; [g]\; \backslash subseteq\; H^\backslash infty\; +\; C^0(S^1)$.

Here, $H^\backslash infty$ denotes the closed subalgebra of $L^\backslash infty\; (S^1)$ of analytic functions (functions with vanishing negative Fourier coefficients), $H^\backslash infty\; [f]$ is the closed subalgebra of $L^\backslash infty\; (S^1)$ generated by $f$ and $H^\backslash infty$, and $C^0(S^1)$ is the space (as an algebraic set) of continuous functions on the circle. See {{harvtxt|S.Axler, S-Y. Chang, D. Sarason |1978}}.

• {{annotated link|Toeplitz matrix}}

{{reflist}}

• {{citation |last= S.Axler, S-Y. Chang, D. Sarason|title=Products of Toeplitz operators|journal=Integral Equations and Operator Theory|volume= 1 |year=1978|issue=3 |pages= 285–309|doi=10.1007/BF01682841 |s2cid=120610368 }}
• {{citation|first1=Albrecht | last1= Böttcher| first2=Sergei M. | last2=Grudsky |title=Toeplitz Matrices, Asymptotic Linear Algebra, and Functional Analysis| url=https://books.google.com/books?id=Dmr0BwAAQBAJ&pg=PA1| year=2000 |publisher=Birkhäuser |isbn=978-3-0348-8395-5}}.
• {{citation|last1=Böttcher|first1=A.|author1-link=Albrecht Böttcher|last2=Silbermann|first2=B.|year=2006|title=Analysis of Toeplitz Operators|edition=2nd|publisher=Springer-Verlag|series=Springer Monographs in Mathematics|isbn= 978-3-540-32434-8}}.
• {{citation|first=Ronald|last=Douglas|authorlink=Ronald Douglas|title=Banach Algebra techniques in Operator theory|publisher=Academic Press|year=1972}}.
• {{citation|first1=Marvin|last1=Rosenblum|first2=James|last2=Rovnyak|title=Hardy Classes and Operator Theory|year=1985|publisher=Oxford University Press}}. Reprinted by Dover Publications, 1997, {{isbn|978-0-486-69536-5}}.

{{DEFAULTSORT:Toeplitz Operator}}

Category:Operator theory

Category:Hardy spaces

Category:Linear operators

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