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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Published in | Mediterranean journal of mathematics Vol. 20; no. 6 |
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Main Authors | , , , |
Format | Journal Article |
Language | English |
Published |
Cham
Springer International Publishing
01.12.2023
Springer Nature B.V |
Subjects | |
Online Access | Get full text |
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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. |
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ISSN: | 1660-5446 1660-5454 |
DOI: | 10.1007/s00009-023-02519-7 |