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Toeplitz operator

## Summary

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

## Details

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

${\displaystyle T_{g}=PM_{g}\vert _{H^{2}},}$

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.

## Theorems

• Theorem: If ${\displaystyle g}$  is continuous, then ${\displaystyle T_{g}-\lambda }$  is Fredholm if and only if ${\displaystyle \lambda }$  is not in the set ${\displaystyle g(S^{1})}$ . If it is Fredholm, its index is minus the winding number of the curve traced out by ${\displaystyle g}$  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.

• Axler- Chang- Sarason Theorem: The operator ${\displaystyle T_{f}T_{g}-T_{fg}}$  is compact if and only if ${\displaystyle H^{\infty }[{\bar {f}}]\cap H^{\infty }[g]\subseteq H^{\infty }+C}$ .

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

## References

• S.Axler, S-Y. Chang, D. Sarason (1978), "Products of Toeplitz operators", Integral Equations and Operator Theory, 1: 285–309{{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.