CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCJM
Abstract view

The Bishop-Phelps-Bollobás property for compact operators

  • 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
Format:   LaTeX   MathJax   PDF  

Abstract

We study the Bishop-Phelps-Bollobás 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 Radon-Nikodým 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 finite-dimensional 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 infinite-dimensional. 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 $K$.
Keywords: Bishop-Phelps theorem, Bishop-Phelps-Bollobás property, norm attaining operator, compact operator Bishop-Phelps theorem, Bishop-Phelps-Bollobás property, norm attaining operator, compact operator
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 operator-valued 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 operator-valued functions
 

© Canadian Mathematical Society, 2017 : https://cms.math.ca/