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

Details

Let S 1 {\displaystyle S^{1}} be the unit circle in the complex plane, with the standard Lebesgue measure, and L 2 ( S 1 ) {\displaystyle L^{2}(S^{1})} be the Hilbert space of complex-valued square-integrable functions. A bounded measurable complex-valued function g {\displaystyle g} on S 1 {\displaystyle S^{1}} defines a multiplication operator M g {\displaystyle M_{g}} on L 2 ( S 1 ) {\displaystyle L^{2}(S^{1})} . Let P {\displaystyle P} be the projection from L 2 ( S 1 ) {\displaystyle L^{2}(S^{1})} onto the Hardy space H 2 {\displaystyle H^{2}}. The Toeplitz operator with symbol g {\displaystyle g} is defined by

T g = P M g | H 2 , {\displaystyle T_{g}=PM_{g}\vert _{H^{2}},}

where " | " means restriction.

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

Theorems

  • Theorem: If g {\displaystyle g} is continuous, then T g − λ {\displaystyle T_{g}-\lambda } is Fredholm if and only if λ {\displaystyle \lambda } is not in the set g ( S 1 ) {\displaystyle g(S^{1})}. If it is Fredholm, its index is minus the winding number of the curve traced out by g {\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 T f T g − T f g {\displaystyle T_{f}T_{g}-T_{fg}} is compact if and only if H ∞ [ f ¯ ] ∩ H ∞ [ g ] ⊆ H ∞ + C 0 ( S 1 ) {\displaystyle H^{\infty }[{\bar {f}}]\cap H^{\infty }[g]\subseteq H^{\infty }+C^{0}(S^{1})}.

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

See also

  • S.Axler, S-Y. Chang, D. Sarason (1978), "Products of Toeplitz operators", Integral Equations and Operator Theory, 1 (3): 285–309, doi:, S2CID{{citation}}: CS1 maint: multiple names: authors list (link)
  • Böttcher, Albrecht; Grudsky, Sergei M. (2000), , 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.