Department of Mathematics

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On convergence of sections of sequences in Banach spaces
(Springer-Verlag, 2000-02-01) Aywa, Shem; Jan, Fourie
An elementary proof of the (known) fact that each element of the Banach spaceℓ w p (X) of weakly absolutelyp-summable sequences (if 1≤p<∞) in the Banach spaceX is the norm limit of its sections if and only if each element ofℓ w p (X) is a norm null sequence inX, is given. Little modification to this proof leads to a similar result for a family of Orlicz sequence spaces. Some applications to spaces of compact operators on Banach sequence spaces are considered.
Spaces of compact operators and their dual spaces
(Springer-Link, 2004-01-13) Aywa, Shem; Jan, Fourie
Theω′-topology on the spaceL(X, Y) of bounded linear operators from the Banach spaceX into the Banach spaceY is discussed in [10]. Let ℒw' (X, Y) denote the space of allT∈L(X, Y) for which there exists a sequence of compact linear operators (T n)⊂K(X, Y) such thatT=ω′−limnTn and let|||T|||:={supn||Tn||:Tn∈K(X,Y),Tn→w′T}. We show that(Lw′,|||⋅|||) is a Banach ideal of operators and that the continuous dual spaceK(X, Y)* is complemented in(Lw′(X,Y),|||⋅|||)∗. This results in necessary and sufficient conditions forK(X, Y) to be reflexive, whereby the spacesX andY need not satisfy the approximation property. Similar results follow whenX andY are locally convex spaces.