Toeplitz operators on the space of real analytic functions: The Fredholm property

Domański, Paweł; Jasiczak, M.

doi:10.1215/17358787-2017-0022

Cited by 5 publications

(17 citation statements)

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“…In order to prove the theorem we again prove a representation of the range of the adjoint operator T F of the form (9). This theorem implies the following analog of Corollary 2.3:…”

Section: Theorem 4 Assume That F ∈ X (R) Has Non-real Zeros Accumulamentioning

confidence: 89%

“…Our work relies heavily on the previous results concerning Fredholm Toeplitz operators [9,27] and semi-Fredholm operators [28]. We also use extensively the Köthe-Grothendieck-da Silva duality.…”

Section: Corollary 24 Consider a Toeplitz Operator T F : A(r) → A(rmentioning

confidence: 99%

“…For more details we refer to our previous works [9,27] and, especially, [28]. We refer the reader to [26,36] and [30] for the functional analytic background.…”

Section: The Space Of Real Analytic Functionsmentioning

confidence: 99%

“…In [32] Langenbruch solved the right inverse problem for convolution operators on the space of real analytic functions. In this paper we consider similar problems for the class of Toeplitz operators, which we introduced in [9]. Our approach is however completely different.…”

Section: Introductionmentioning

confidence: 99%

“…We say that the matrix M T is associated with the operator T . It was proved in ( [9], Theorem 1) that if the matrix associated with a continuous linear operator T : A(R) → A(R) is a Toeplitz matrix, then T is a Toeplitz operator. That is, T is necessarily an operator T F defined in (6) for some symbol F ∈ X (R).…”

Section: Introductionmentioning

confidence: 99%

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

Jasiczak

Golińska

2020

Integr. Equ. Oper. Theory

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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.

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“…In order to prove the theorem we again prove a representation of the range of the adjoint operator T F of the form (9). This theorem implies the following analog of Corollary 2.3:…”

Section: Theorem 4 Assume That F ∈ X (R) Has Non-real Zeros Accumulamentioning

confidence: 89%

Section: Corollary 24 Consider a Toeplitz Operator T F : A(r) → A(rmentioning

confidence: 99%

“…For more details we refer to our previous works [9,27] and, especially, [28]. We refer the reader to [26,36] and [30] for the functional analytic background.…”

Section: The Space Of Real Analytic Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations