# Spear Operators Between Banach Spaces (lecture Notes In Mathematics)

by Vladimir Kadets /
2018 / English / PDF

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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

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
$\|G + \omega\,T\|=1+ \|T\|$.

$\|G + \omega\,T\|=1+ \|T\|$.
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.

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.

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.