Compact operators whose adjoints factor through subspaces of $l_p$

Deba P. Sinha, and Anil K. Karn. "Compact operators whose adjoints factor through subspaces of $l_{p}$." Studia Mathematica 150.1 (2002): 17-33. <http://eudml.org/doc/286285>.

@article{DebaP2002,
abstract = {For p ≥ 1, a subset K of a Banach space X is said to be relatively p-compact if $ K ⊂ \{∑_\{n=1\}^\{∞\} αₙxₙ: \{αₙ\} ∈ Ball(l_\{p^\{\prime \}\})\}$, where p’ = p/(p-1) and $\{xₙ\} ∈ l_\{p\}^\{s\}(X)$. An operator T ∈ B(X,Y) is said to be p-compact if T(Ball(X)) is relatively p-compact in Y. Similarly, weak p-compactness may be defined by considering $\{xₙ\} ∈ l_\{p\}^\{w\}(X)$. It is proved that T is (weakly) p-compact if and only if T* factors through a subspace of $l_\{p\}$ in a particular manner. The normed operator ideals $(K_\{p\},κ_\{p\})$ of p-compact operators and $(W_\{p\},ω_\{p\})$ of weakly p-compact operators, arising from these factorizations, are shown to be complete. It is also shown that the adjoints of p-compact operators are p-summing. It is further proved that for p ≥ 1 the identity operator on X can be approximated uniformly on every p-compact set by finite rank operators, or in other words, X has the p-approximation property, if and only if for every Banach space Y the set of finite rank operators is dense in the ideal $K_\{p\}(Y,X)$ of p-compact operators in the factorization norm $ω_\{p\}$. It is also proved that every Banach space has the 2-approximation property while for each p > 2 there is a Banach space that fails the p-approximation property.},
author = {Deba P. Sinha, Anil K. Karn},
journal = {Studia Mathematica},
keywords = {approximation property; -compact operator; -compact set; factorization of operators; operator ideal space},
language = {eng},
number = {1},
pages = {17-33},
title = {Compact operators whose adjoints factor through subspaces of $l_\{p\}$},
url = {http://eudml.org/doc/286285},
volume = {150},
year = {2002},
}

TY - JOUR
AU - Deba P. Sinha
AU - Anil K. Karn
TI - Compact operators whose adjoints factor through subspaces of $l_{p}$
JO - Studia Mathematica
PY - 2002
VL - 150
IS - 1
SP - 17
EP - 33
AB - For p ≥ 1, a subset K of a Banach space X is said to be relatively p-compact if $ K ⊂ {∑_{n=1}^{∞} αₙxₙ: {αₙ} ∈ Ball(l_{p^{\prime }})}$, where p’ = p/(p-1) and ${xₙ} ∈ l_{p}^{s}(X)$. An operator T ∈ B(X,Y) is said to be p-compact if T(Ball(X)) is relatively p-compact in Y. Similarly, weak p-compactness may be defined by considering ${xₙ} ∈ l_{p}^{w}(X)$. It is proved that T is (weakly) p-compact if and only if T* factors through a subspace of $l_{p}$ in a particular manner. The normed operator ideals $(K_{p},κ_{p})$ of p-compact operators and $(W_{p},ω_{p})$ of weakly p-compact operators, arising from these factorizations, are shown to be complete. It is also shown that the adjoints of p-compact operators are p-summing. It is further proved that for p ≥ 1 the identity operator on X can be approximated uniformly on every p-compact set by finite rank operators, or in other words, X has the p-approximation property, if and only if for every Banach space Y the set of finite rank operators is dense in the ideal $K_{p}(Y,X)$ of p-compact operators in the factorization norm $ω_{p}$. It is also proved that every Banach space has the 2-approximation property while for each p > 2 there is a Banach space that fails the p-approximation property.
LA - eng
KW - approximation property; -compact operator; -compact set; factorization of operators; operator ideal space
UR - http://eudml.org/doc/286285
ER -