Unconditional banach space ideal property

Abstract

Let Lw ′ denote the assignment which associates with each pair of Banach spaces X , Y , the vector space Lw ′ ( X , Y ) and K ( X , Y ) be the space of all compact linear operators from X to Y. Let T ∈ Lw ′ ( X , Y ) and suppose (Tn ) ⊂ K ( X , Y ) converges in the dual weak operator topology (w′) of T. Denote by K u ((Tn )) the finite number given by K u ((Tn )) := sup { max { Tn , T − 2Tn }} . n∈N ′ The u-norm on Lw ( X , Y ) is then given by T u := inf { K u ((Tn )) : T = w′ − lim Tn , n Tn ∈ K ( X , Y )}. ′ It has been shown that ( Lw ( X , Y ) . u ) is a Banach operator ideal. We find ′ conditions for K ( X , Y ) to be an unconditional ideal in ( Lw ( X , Y ) . u ) .