A new approach on Lipschitz compact operators

Cabrera-Padilla, M. G.; Jiménez-Vargas, A.

doi:10.1016/j.topol.2015.12.072

Cited by 7 publications

(7 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The first result which we present below will be very useful in the sequel. It is the straightforward extension of [11,Theorem 2.3]. Proposition 4.1.…”

Section: Compact and Weakly Compact Weighted Operatorsmentioning

confidence: 96%

“…The second lemma is an adaptation of [11,Theorem 2.3] for K = C. In the statement below, we let For µ ∈ acoM, write f, µ = re iθ . Note that e −iθ µ ∈ acoM and f, e…”

Section: Now the Conclusion Follows From The Basic Inequalitiesmentioning

confidence: 99%

“…Section 4 deals with compactness and weak compactness of weighted Lipschitz operators and weighted composition operators. A key point is the equivalent wording provided in Proposition 4.1, which in turns is an adaptation of [11,Theorem 2.3]. Another important ingredient is Theorem 2.14 which allows us to prove that weak compactness is actually equivalent to (norm) compactness for the class of operators we consider in this article.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the Dynamics of Lipschitz Operators

Abbar

Coine

Petitjean

2021

Integr. Equ. Oper. Theory

View full text Add to dashboard Cite

We consider weighted composition operators, that is operators of the type g → w • g • f , acting on spaces of Lipschitz functions. Bounded weighted composition operators, as well as some compact weighted composition operators, have been characterized quite recently. In this paper, we provide a different approach involving their pre-adjoint operators, namely the weighted Lipschitz operators acting on Lipschitz free spaces. This angle allows us to improve some results from the literature. Notably, we obtain a distinct characterization of boundedness with a precise estimate of the norm. We also characterise injectivity, surjectivity, compactness and weak compactness in full generality.

show abstract

“…The first result which we present below will be very useful in the sequel. It is the straightforward extension of [11,Theorem 2.3]. Proposition 4.1.…”

Section: Compact and Weakly Compact Weighted Operatorsmentioning

confidence: 96%

“…The second lemma is an adaptation of [11,Theorem 2.3] for K = C. In the statement below, we let For µ ∈ acoM, write f, µ = re iθ . Note that e −iθ µ ∈ acoM and f, e…”

Section: Now the Conclusion Follows From The Basic Inequalitiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the Dynamics of Lipschitz Operators

Abbar

Coine

Petitjean

2021

Integr. Equ. Oper. Theory

View full text Add to dashboard Cite

show abstract

“…The main objective of this section is to prove theorem A. The proof will be based on the next easy but smart observation from [11] (see theorem 2.3 therein). This result concerns not only compact operators but also weakly compact operators, and so it will be useful in § 3 as well.…”

Section: A Metric Characterization Of Compact Lipschitz Operatorsmentioning

confidence: 99%

Compact and weakly compact Lipschitz operators

Abbar¹,

Coine²,

Petitjean³

2022

Proceedings of the Royal Society of Edinburgh: Section A Mathem

View full text Add to dashboard Cite

Any Lipschitz map $f : M \to N$ between two pointed metric spaces may be extended in a unique way to a bounded linear operator $\widehat {f} : \mathcal {F}(M) \to \mathcal {F}(N)$ between their corresponding Lipschitz-free spaces. In this paper, we give a necessary and sufficient condition for $\widehat {f}$ to be compact in terms of metric conditions on $f$ . This extends a result by A. Jiménez-Vargas and M. Villegas-Vallecillos in the case of non-separable and unbounded metric spaces. After studying the behaviour of weakly convergent sequences made of finitely supported elements in Lipschitz-free spaces, we also deduce that $\widehat {f}$ is compact if and only if it is weakly compact.

show abstract

“…The main objective of this section is to prove Theorem A. The proof will be based on the next easy but smart observation from [10] (see Theorem 2.3 therein). This result concerns not only compact operators but also weakly compact operators, and so it will be useful in Section 3 as well.…”

Section: A Metric Characterisation Of Compact Lipschitz Operatorsmentioning

confidence: 99%

Compact and weakly compact Lipschitz operators

Abbar¹,

Coine²,

Petitjean³

2021

Preprint

View full text Add to dashboard Cite

Any Lipschitz map f : M → N between two pointed metric spaces may be extended in a unique way to a bounded linear operator f : F (M ) → F (N ) between their corresponding Lipschitz-free spaces. In this paper, we give a necessary and sufficient condition for f to be compact in terms of metric conditions on f . This extends a result by A. Jiménez-Vargas and M. Villegas-Vallecillos in the case of non-separable and unbounded metric spaces. After studying the behavior of weakly convergent sequences made of finitely supported elements in Lipschitz-free spaces, we also deduce that f is compact if and only if it is weakly compact.

show abstract

A new approach on Lipschitz compact operators

Cited by 7 publications

References 13 publications

On the Dynamics of Lipschitz Operators

On the Dynamics of Lipschitz Operators

Compact and weakly compact Lipschitz operators

Compact and weakly compact Lipschitz operators

Contact Info

Product

Resources

About