Abstract
We show that a Toeplitz operator on the space of real analytic functions on the real line is left invertible if and only if it is an injective Fredholm operator, it is right invertible if and only if it is a surjective Fredholm operator. The characterizations are given in terms of the winding number of the symbol of the operator. Our results imply that the range of a Toeplitz operator (and also its adjoint) is complemented if and only if it is of finite codimension. Similarly, the kernel of a Toeplitz operator (and also its adjoint) is complemented if and only if it is of finite dimension.
Highlights
In this paper we study Toeplitz operators on the space of real analytic functions on the real line
Some holomorphic functions develop into Newton series a0 +
Is well-defined and formula (4) holds true for functions which develop into power series. This is why we consider in this paper the case of real analytic functions
Summary
In this paper we study Toeplitz operators on the space of real analytic functions on the real line. We start by presenting the classical background for our investigation. The presentation is based on the beautiful books by Gelfond [21] and Markushevich [34]. A fundamental result in function theory says that a holomorphic function locally develops into a power series. There are other series representations of holomorphic functions. Some holomorphic functions develop into Newton series a0 +.
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