Spear Operators Between Banach Spaces

Description

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 $T:X\longrightarrow Y$ there exists a modulus-one scalar $\omega$ such that$\|G + \omega\,T\|=1+ \|T\|$. 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 $L_1$. 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 interest 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. Front Matter ....Pages i-xv Historical Introduction: A Walk on the Results for Banach Spaces with Numerical Index 1 (Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez)....Pages 1-36 Spear Vectors and Spear Sets (Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez)....Pages 37-47 Three Definitions for Operators: Spearness, the Alternative Daugavet Property, and Lushness (Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez)....Pages 49-66 Some Examples in Classical Banach Spaces (Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez)....Pages 67-82 Further Results (Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez)....Pages 83-95 Isometric and Isomorphic Consequences (Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez)....Pages 97-102 Lipschitz Spear Operators (Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez)....Pages 103-113 Some Stability Results (Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez)....Pages 115-150 Open Problems (Vladimir Kadets, Miguel Martín, Javier Merí, Antonio Pérez)....Pages 151-152 Back Matter ....Pages 153-164