Non-rough Norms and Dentability in Spaces of Operators

In this work, we study non-rough norms in L ( X ,  Y ),  the space of bounded linear operators between Banach spaces X and Y . We prove that L ( X ,  Y ) has non-rough norm if and only if X ∗ and Y have non-rough norm. We show that the injective tensor product X ⊗ ^ ε Y has non-rough norm if and onl...

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Bibliographic Details
Published inMediterranean journal of mathematics Vol. 20; no. 6
Main Authors Basu, Sudeshna, Guerrero, Julio Becerra, Seal, Susmita, Yeguas, Juan Miguel Villegas
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.12.2023
Springer Nature B.V
Subjects
Banach spaces
Linear operators
Mathematical analysis
Mathematics
Mathematics and Statistics
Norms
Operators (mathematics)
Tensors
46B28
dentable
small combination of slices
Slices
denting
46B20
space of operators
huskable
stability
tensor product
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Summary:In this work, we study non-rough norms in L ( X ,  Y ),  the space of bounded linear operators between Banach spaces X and Y . We prove that L ( X ,  Y ) has non-rough norm if and only if X ∗ and Y have non-rough norm. We show that the injective tensor product X ⊗ ^ ε Y has non-rough norm if and only if both X and Y have non-rough norm. We also give an example to show that non-rough norms are not stable under projective tensor product. We also study a related concept namely the small diameter properties in the context of L ( X , Y ) ∗ . These results lead to a discussion on stability of the small diameter properties for projective and injective tensor product spaces.
ISSN:1660-5446
1660-5454
DOI:10.1007/s00009-023-02519-7