Norm-attaining operators which satisfy a Bollobas type theorem

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F21%3A00354990" target="_blank" >RIV/68407700:21230/21:00354990 - isvavai.cz</a>

Result on the web

<a href="https://doi.org/10.1007/s43037-020-00113-7" target="_blank" >https://doi.org/10.1007/s43037-020-00113-7</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s43037-020-00113-7" target="_blank" >10.1007/s43037-020-00113-7</a>

Result language

angličtina

Original language name

Norm-attaining operators which satisfy a Bollobas type theorem

Original language description

In this paper, we are interested in studying the set A(parallel to center dot parallel to) (X, Y) of all norm-attaining operators T from X into Y satisfying the following: given epsilon > 0, there exists eta such that if parallel to Tx parallel to > 1 - eta, then there is x(0) such that parallel to x(0) - x parallel to < epsilon and T itself attains its norm at x(0). We show that every norm one functional on c(0) which attains its norm belongs to A(parallel to center dot parallel to) (c(0), K). Also, we prove that the analogous result holds neither for A(parallel to center dot parallel to) (l(1), K) nor A(parallel to center dot parallel to) (l(infinity), K). Under some assumptions, we show that the sphere of the compact operators belongs to A(parallel to center dot parallel to) (X, Y) and that this is no longer true when some of these hypotheses are dropped. The analogous set A(nu)(X) for numerical radius of an operator instead of its norm is also defined and studied. We present a complete characterization for the diagonal operators which belong to the sets A(parallel to center dot parallel to) (X, X) and A(nu)(X) when X = c(0) or l(p). As a consequence, we get that the canonical projections P-N on these spaces belong to our sets. We give examples of operators on infinite dimensional Banach spaces which belong to A(parallel to center dot parallel to) (X, X) but not to A(nu)(X) and vice-versa. Finally, we establish some techniques which allow us to connect both sets by using direct sums.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

10101 - Pure mathematics

Project

<a href="/en/project/EF16_019%2F0000778" target="_blank" >EF16_019/0000778: Center for advanced applied science</a><br>

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2021

Confidentiality

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

Name of the periodical

Banach Journal of Mathematical Analysis

ISSN

2662-2033

e-ISSN

1735-8787

Volume of the periodical

15

Issue of the periodical within the volume

2

Country of publishing house

CH - SWITZERLAND

Number of pages

26

Pages from-to

1-26

UT code for WoS article

000627791600001

EID of the result in the Scopus database

2-s2.0-85102415681