Generating operators between Banach spaces

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21240%2F24%3A00377163" target="_blank" >RIV/68407700:21240/24:00377163 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1007/s13398-024-01582-3" target="_blank" >https://doi.org/10.1007/s13398-024-01582-3</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s13398-024-01582-3" target="_blank" >10.1007/s13398-024-01582-3</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Generating operators between Banach spaces

Popis výsledku v původním jazyce

We introduce and study the notion of generating operators as those norm-one operators G:X⟶Y such that for every 0<δ<1, the set {xelementX:‖x‖⩽1,‖Gx‖>1-δ} generates the unit ball of X by closed convex hull. This class of operators includes isometric embeddings, spear operators (actually, operators with the alternative Daugavet property), and other examples like the natural inclusions of ℓ1 into c0 and of Linfinity[0,1] into L1[0,1]. We first present a characterization in terms of the adjoint operator, make a discussion on the behaviour of diagonal generating operators on c0-, ℓ1-, and ℓinfinity-sums, and present examples in some classical Banach spaces. Even though rank-one generating operators always attain their norm, there are generating operators, even of rank-two, which do not attain their norm. We discuss when a Banach space can be the domain of a generating operator which does not attain its norm in terms of the behaviour of some spear sets of the dual space. Finally, we study when the set of all generating operators between two Banach spaces X and Y generates all non-expansive operators by closed convex hull. We show that this is the case when X=L1(μ) and Y has the Radon-Nikodým property with respect to μ. Therefore, when X=ℓ1(Γ), this is the case for every target space Y. Conversely, we also show that a real finite-dimensional space X satisfies that generating operators from X to Y generate all non-expansive operators by closed convex hull only in the case that X is an ℓ1-space.

Název v anglickém jazyce

Generating operators between Banach spaces

Popis výsledku anglicky

We introduce and study the notion of generating operators as those norm-one operators G:X⟶Y such that for every 0<δ<1, the set {xelementX:‖x‖⩽1,‖Gx‖>1-δ} generates the unit ball of X by closed convex hull. This class of operators includes isometric embeddings, spear operators (actually, operators with the alternative Daugavet property), and other examples like the natural inclusions of ℓ1 into c0 and of Linfinity[0,1] into L1[0,1]. We first present a characterization in terms of the adjoint operator, make a discussion on the behaviour of diagonal generating operators on c0-, ℓ1-, and ℓinfinity-sums, and present examples in some classical Banach spaces. Even though rank-one generating operators always attain their norm, there are generating operators, even of rank-two, which do not attain their norm. We discuss when a Banach space can be the domain of a generating operator which does not attain its norm in terms of the behaviour of some spear sets of the dual space. Finally, we study when the set of all generating operators between two Banach spaces X and Y generates all non-expansive operators by closed convex hull. We show that this is the case when X=L1(μ) and Y has the Radon-Nikodým property with respect to μ. Therefore, when X=ℓ1(Γ), this is the case for every target space Y. Conversely, we also show that a real finite-dimensional space X satisfies that generating operators from X to Y generate all non-expansive operators by closed convex hull only in the case that X is an ℓ1-space.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2024

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas

ISSN

1578-7303

e-ISSN

1579-1505

Svazek periodika

118

Číslo periodika v rámci svazku

3

Stát vydavatele periodika

DE - Spolková republika Německo

Počet stran výsledku

25

Strana od-do

—

Kód UT WoS článku

001202051900002

EID výsledku v databázi Scopus

2-s2.0-85190373899