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Toeplitz operators on the space of all entire functions. (English) Zbl 1512.47054

Toeplitz operators on the Fréchet space \(H(\mathbb{C})\) of entire functions are introduced and analyzed. These operators are defined as linear continuous operators on \(H(\mathbb{C})\) with Toeplitz matrices with respect to the basis \(\lbrace z^n\rbrace_{n\ge0}\). The author obtains a representation of such operators as a product of a multiplication operator by an entire function and some Cauchy transform. The symbol space of these operators is the space \(H(\mathbb{C})\oplus H_0(\infty)\) where \(H_0(\infty)\) is the space of all germs at \(\infty\) of holomorphic functions vanishing at this point. The author characterizes Fredholm, semi-Fredholm and one-sided invertible operators from this class and determines the index of the Fredholm operators.

MSC:

47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
30H99 Spaces and algebras of analytic functions of one complex variable
47A53 (Semi-) Fredholm operators; index theories
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