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Spear operators between Banach spaces

Author: V M Kadet︠s︡; Miguel Martín; Javier Merí; Antonio Pérez
Publisher: Cham : Springer, [2018] ©2018
Series: Lecture notes in mathematics (Springer-Verlag), 2205.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:
"This monograph is devoted to the study of spear operators, that is, bounded linear operators G between Banach spaces X and Y satisfying that for every other bounded linear operator T : X --> Y there exists a modulus-one scalar w such that ||G + w T|| = 1 + ||T||. This concept extends the properties of the identity operator in those Banach spaces having numerical index one. Many examples among classical spaces  Read more...
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Document Type: Book
All Authors / Contributors: V M Kadet︠s︡; Miguel Martín; Javier Merí; Antonio Pérez
ISBN: 3319713329 9783319713328
OCLC Number: 1007087956
Description: xv, 161 pages : illustrations ; 24 cm.
Contents: Historical introduction : a walk on the results for Banach spaces with numerical index 1 --
Spear vectors and spear sets --
Three definitions for operators : spearness, the alternative daugavet property, and lushness --
Some examples in classical Banach spaces --
Further results --
Isometric and isomorphic consequences --
Lipschitz spear operators --
Some stability results --
Open problems.
Series Title: Lecture notes in mathematics (Springer-Verlag), 2205.
Responsibility: Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez.

Abstract:

"This monograph is devoted to the study of spear operators, that is, bounded linear operators G between Banach spaces X and Y satisfying that for every other bounded linear operator T : X --> Y there exists a modulus-one scalar w such that ||G + w T|| = 1 + ||T||. This concept extends the properties of the identity operator in those Banach spaces having numerical index one. Many examples among classical spaces are provided, being one of them the Fourier transform on L₁. The relationships with the Radon-Nikodým property, with Asplund spaces and with the duality, and some isometric and isomorphic consequences are provided. Finally, Lipschitz operators satisfying the Lipschitz version of the equation above are studied. The book could be of interests to young researchers and specialists in functional analysis, in particular to those interested in Banach spaces and their geometry. It is essentially self-contained and only basic knowledge of functional analysis is needed"--Page 4 of cover.

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