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 ) .
