Generating operators between Banach spaces
The result's identifiers
Result code in 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>
Result on the web
<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>
Alternative languages
Result language
angličtina
Original language name
Generating operators between Banach spaces
Original language description
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.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2024
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas
ISSN
1578-7303
e-ISSN
1579-1505
Volume of the periodical
118
Issue of the periodical within the volume
3
Country of publishing house
DE - GERMANY
Number of pages
25
Pages from-to
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UT code for WoS article
001202051900002
EID of the result in the Scopus database
2-s2.0-85190373899