Compact operators which factor through subspaces of lp

✍ Scribed by Deba Prasad Sinha; Anil Kumar Karn

• Publisher: John Wiley and Sons
• Year: 2008
• Tongue: English
• Weight: 194 KB
• Volume: 281
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200510612

✦ Synopsis

Abstract

Let 1 ≤ p ≤ ∞. A subset K of a Banach space X is said to be relatively p ‐compact if there is an 〈x~n~ 〉 ∈ l^s^ ~p~ (X) such that for every k ∈ K there is an 〈α~n~ 〉 ∈ l~p ′~ such that k = σ^∞^~n=1~ α~n~ x~n~ . A linear operator T: X → Y is said to be p ‐compact if T (Ball (X)) is relatively p ‐compact in Y. The set of all p ‐compact operators K~p~ (X, Y) from X to Y is a Banach space with a suitable factorization norm κ~p~ and (K~p~ , κ~p~ ) is a Banach operator ideal. In this paper we investigate the dual operator ideal (K^d^ ~p~ , κ^d^ ~p~ ). It is shown that κ^d^ ~p~ (T) = π~p~ (T) for all T ∈ B (X, Y) if either X or Y is finite‐dimensional. As a consequence it is proved that the adjoint ideal of K^d^ ~p~ is I~p ′~, the ideal of p ′‐integral operators. Further, a composition/decomposition theorem K^d^ ~p~ = Π~p~ K is proved which also yields that (Π^min^ ~p~ )^inj^ = K^d^ ~p~ . Finally, we discuss the density of finite rank operators in K^d^ ~p~ and give some examples for different values of p in this respect. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)