One-Sided Invertibility of Toeplitz Operators on the Space of Real Analytic Functions on the Real Line

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.

Toeplitz operators on the space of all holomorphic functions on finitely connected domains

“…It is just one step from the operators, which are in some sense diagonal, to the operators, the associated matrices of which are Toeplitz and Hankel. We made this step in [18] and also in [28][29][30] and investigated Toeplitz operators on the space of all real analytic functions on the real line A(R). Golińska in [25] studied the Hankel case.…”

“…The operator P ∞ extends to a continuous projection on S ∞ onto H (C) along H 0 (∞). The formula for the extension is again (28) with R > 0 large enough.…”

“…For such a domain D let us also consider the space X = H (D). Let the symbol be equal to F ⊕ F ∞ with F ∞ not equal to zero (see formula (30) in Sect. 4).…”

“…We study Toeplitz operators on the space of all holomorphic functions on finitely connected domains in C or to put it in another way on finitely connected submanifolds of the Riemann sphere with boundary. We introduced and investigated the class of Toeplitz operators on the space of all real analytic functions on the real line in [18,[28][29][30]. The methods developed therein turned out to be general and universal enough to be applied to the other spaces of holomorphic functions.…”

“…They are related to Newton series, close cousins of Dirichlet series. We will not repeat these applications here and refer the reader to [28,31] and especially to [30,Introduction] for an explanation. The theory of continuous linear operators on locally convex spaces, including the spaces of analytic functions, is much less developed than the Banach space counterpart, not to mention the Hilbert one.…”