For $p\geq 1$, a set $K$ in a Banach space $X$ is said to be relatively $p$-compact if there exists a $p$-summable sequence $(x_n)$ in $X$ with $K\subseteq
\{\sum_n\alpha_nx_n : (\alpha_n)\in B_{\ell_{p'}}\}$. We prove that an operator $T\colon X\rightarrow Y$ is $p$-compact (i.e., $T$ maps bounded sets to relatively $p$-compact sets) iff $T^*$ is quasi $p$-nuclear. Further, we characterize $p$-summing operators as those operators whose adjoints map relatively compact sets to relatively $p$-compact sets.
Keywords:
geq set banach space said relatively p compact there exists p summable sequence subseteq sum alpha alpha ell prove operator colon rightarrow p compact maps bounded sets relatively p compact sets * quasi p nuclear further characterize p summing operators those operators whose adjoints map relatively compact sets relatively p compact sets
Affiliations des auteurs :
J. M. Delgado
1
;
C. Piñeiro
1
;
E. Serrano
1
1
Departamento de Matemáticas Campus Universitario del Carmen Universidad de Huelva Avda. de las Fuerzas Armadas s//n 21071 Huelva, Spain
@article{10_4064_sm197_3_6,
author = {J. M. Delgado and C. Pi\~neiro and E. Serrano},
title = {Operators whose adjoints are quasi $p$-nuclear},
journal = {Studia Mathematica},
pages = {291--304},
year = {2010},
volume = {197},
number = {3},
doi = {10.4064/sm197-3-6},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/sm197-3-6/}
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AU - E. Serrano
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J. M. Delgado; C. Piñeiro; E. Serrano. Operators whose adjoints are quasi $p$-nuclear. Studia Mathematica, Tome 197 (2010) no. 3, pp. 291-304. doi: 10.4064/sm197-3-6
