Quantification of Pelczynski's property (V)
Result description
A Banach space X has Pelczynski's property (V) if for every Banach space Y every unconditionally converging operator T : X -> Y is weakly compact. In 1962, Aleksander Pelczynski showed that C(K) spaces for a compact Hausdorff space K enjoy the property (V), and some generalizations of this theorem have been proved since then. We introduce several possibilities of quantifying the property (V). We prove some characterizations of the introduced quantitative versions of this property, which allow us to prove a quantitative version of Pelczynski's result about C(K) spaces and generalize it. Finally, we study the relationship of several properties of operators including weak compactness and unconditional convergence, and using the results obtained we establish a relation between quantitative versions of the property (V) and quantitative versions of other well known properties of Banach spaces.
Keywords
unconditionally converging operatorsquantitative Pelczynski's property ( V)Pelczynski's property (V)Measures of weak non-compactness
The result's identifiers
Result code in IS VaVaI
Result on the web
DOI - Digital Object Identifier
Alternative languages
Result language
angličtina
Original language name
Quantification of Pelczynski's property (V)
Original language description
A Banach space X has Pelczynski's property (V) if for every Banach space Y every unconditionally converging operator T : X -> Y is weakly compact. In 1962, Aleksander Pelczynski showed that C(K) spaces for a compact Hausdorff space K enjoy the property (V), and some generalizations of this theorem have been proved since then. We introduce several possibilities of quantifying the property (V). We prove some characterizations of the introduced quantitative versions of this property, which allow us to prove a quantitative version of Pelczynski's result about C(K) spaces and generalize it. Finally, we study the relationship of several properties of operators including weak compactness and unconditional convergence, and using the results obtained we establish a relation between quantitative versions of the property (V) and quantitative versions of other well known properties of Banach spaces.
Czech name
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Czech description
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Classification
Type
Jimp - 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
GAP201/12/0290: Topological and geometrical properties of Banach spaces and operator algebras
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2017
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Mathematische Nachrichten
ISSN
0025-584X
e-ISSN
—
Volume of the periodical
290
Issue of the periodical within the volume
17-18
Country of publishing house
DE - GERMANY
Number of pages
16
Pages from-to
2909-2924
UT code for WoS article
000419959100017
EID of the result in the Scopus database
2-s2.0-85029223732
Basic information
Result type
Jimp - Article in a specialist periodical, which is included in the Web of Science database
OECD FORD
Pure mathematics
Year of implementation
2017