Abstract view
The BishopPhelpsBollobás property for compact operators


Published:20170525
Printed: Feb 2018
Sheldon Dantas,
Departamento de Análisis Matemático, Universidad de Valencia, Doctor Moliner 50, 46100 Burjasot (Valencia), Spain
Domingo García,
Departamento de Análisis Matemático, Universidad de Valencia, Doctor Moliner 50, 46100 Burjasot (Valencia), Spain
Manuel Maestre,
Departamento de Análisis Matemático, Universidad de Valencia, Doctor Moliner 50, 46100 Burjasot (Valencia), Spain
Miguel Martín,
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain
Abstract
We study the BishopPhelpsBollobás property (BPBp for short)
for compact operators. We present some abstract techniques which
allows to carry the BPBp for compact operators from sequence
spaces to function spaces. As main applications, we prove the
following results. Let $X$, $Y$ be Banach spaces. If $(c_0,Y)$
has the BPBp for compact operators, then so do $(C_0(L),Y)$ for
every locally compact Hausdorff topological space $L$ and $(X,Y)$
whenever $X^*$ is isometrically isomorphic to $\ell_1$.
If $X^*$ has the RadonNikodým property and $(\ell_1(X),Y)$
has the BPBp for compact operators, then so does $(L_1(\mu,X),Y)$
for every positive measure $\mu$; as a consequence, $(L_1(\mu,X),Y)$
has the the BPBp for compact operators when $X$ and $Y$ are finitedimensional
or $Y$ is a Hilbert space and $X=c_0$ or $X=L_p(\nu)$ for any
positive measure $\nu$ and $1\lt p\lt \infty$.
For $1\leq p \lt \infty$, if $(X,\ell_p(Y))$ has the BPBp for compact
operators, then so does $(X,L_p(\mu,Y))$ for every positive measure
$\mu$ such that $L_1(\mu)$ is infinitedimensional. If $(X,Y)$
has the BPBp for compact operators, then so do $(X,L_\infty(\mu,Y))$
for every $\sigma$finite positive measure $\mu$ and $(X,C(K,Y))$
for every compact Hausdorff topological space $K$.
MSC Classifications: 
46B04, 46B20, 46B28, 46B25, 46E40 show english descriptions
Isometric theory of Banach spaces Geometry and structure of normed linear spaces Spaces of operators; tensor products; approximation properties [See also 46A32, 46M05, 47L05, 47L20] Classical Banach spaces in the general theory Spaces of vector and operatorvalued functions
46B04  Isometric theory of Banach spaces 46B20  Geometry and structure of normed linear spaces 46B28  Spaces of operators; tensor products; approximation properties [See also 46A32, 46M05, 47L05, 47L20] 46B25  Classical Banach spaces in the general theory 46E40  Spaces of vector and operatorvalued functions
