Toeplitz operator

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

In operator theory, a Toeplitz operator is the compression of a multiplication operator on the circle to the Hardy space. ==Details== 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 \vert_{H^2}, where " | " means restriction. A bounded operator on H^2 is Toeplitz if and only if its matrix representation, in the basis \{z^n, z \in \mathbb{C}, n \geq 0\} , has constant diagonals. ==Theorems== Theorem: If g is continuous, then T_g - \lambda is Fredholm if and only if \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 . 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^\infty[\bar f] \cap H^\infty [g] \subseteq H^\infty + C^0(S^1) . Here, H^\infty denotes the closed subalgebra of L^\infty (S^1) of analytic functions (functions with vanishing negative Fourier coefficients), H^\infty [f] is the closed subalgebra of L^\infty (S^1) generated by f and H^\infty , and C^0(S^1) is the space (as an algebraic set) of continuous functions on the circle. See . ==See also== ==References== . . . . Reprinted by Dover Publications, 1997, .

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