A numerical range approach to Birkhoff–James orthogonality with applications

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21240%2F24%3A00377162" target="_blank" >RIV/68407700:21240/24:00377162 - isvavai.cz</a>

Result on the web

<a href="https://doi.org/10.1007/s43037-024-00333-1" target="_blank" >https://doi.org/10.1007/s43037-024-00333-1</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s43037-024-00333-1" target="_blank" >10.1007/s43037-024-00333-1</a>

Result language

angličtina

Original language name

A numerical range approach to Birkhoff–James orthogonality with applications

Original language description

The main aim of this paper is to provide characterizations of Birkhoff–James orthogonality (BJ-orthogonality in short) in a number of families of Banach spaces in terms of the elements of significant subsets of the unit ball of their dual spaces, which makes the characterizations more applicable. The tool to do so is a fine study of the abstract numerical range and its relation with the BJ-orthogonality. Among other results, we provide a characterization of BJ-orthogonality for spaces of vector-valued bounded functions in terms of the domain set and the dual of the target space, which is applied to get results for spaces of vector-valued continuous functions, uniform algebras, Lipschitz maps, injective tensor products, bounded linear operators with respect to the operator norm and to the numerical radius, multilinear maps, and polynomials. Next, we study possible extensions of the well-known Bhatia–Šemrl theorem on BJ-orthogonality of matrices, showing results in spaces of vector-valued continuous functions, compact linear operators on reflexive spaces, and finite Blaschke products. Finally, we find applications of our results to the study of spear vectors and spear operators. We show that no smooth point of a Banach space can be BJ-orthogonal to a spear vector of Z. As a consequence, if X is a Banach space containing strongly exposed points and Y is a smooth Banach space with dimension at least two, then there are no spear operators from X to Y. Particularizing this result to the identity operator, we show that a smooth Banach space containing strongly exposed points has numerical index strictly smaller than one. These latter results partially solve some open problems.

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

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OECD FORD branch

10101 - Pure mathematics

Project

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Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2024

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

18

Issue of the periodical within the volume

2

Country of publishing house

CH - SWITZERLAND

Number of pages

35

Pages from-to

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UT code for WoS article

001190556100002

EID of the result in the Scopus database

2-s2.0-85188308373