Browsing by Author "Jan, Fourie"

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Equivalent Banach Operator Ideal Norms1
(HIKARI: International Journal of Mathematical Analysis, 2012-01-01) Musundi, Sammy; Aywa, Shem; Jan, Fourie
Let X, Y be Banach spaces and consider the w′-topology (the dualweak operator topology) on the space (L(X, Y),‖.‖) of bounded linearoperators from X into X with the uniform operator norm.Lw′(X,Y) is the space of all T∈L(X, Y) for which there exists a sequence ofcompact linear operators (Tn)⊂K (X, Y) such thatT=w′−limnTn Two equivalent norms, ‖|T‖|:=inf{█(sup@n)┤‖Tn‖:Tn∈K(X,Y),Tnw′→T}and ‖T‖u:=inf{█(sup@n)┤ {max{‖Tn‖,‖T−2Tn‖}}:‖:Tn∈K(X,Y),Tnw→T}on Lw′(X, Y), are considered. We show that (Lw′,|‖.‖|) and (Lw′,‖.‖u) are Banach operator ideals.
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.
On summing multipliers and applications
(Journal of Mathematical Analysis and Applications, 2001-01-01) Aywa, Shem; Jan, Fourie
A scalar sequence (αi) is said to be a p-summing multiplier of a Banach space E, if ∑∞i = 1‖αixi‖p < ∞ for all weakly p-summable sequences in E. We study some important properties of the space mp(E) of all p-summing multipliers of E, consider applications to E-valued operators on the sequence space lp, and extend this work to general “summing multipliers.” The case p = 1 shows close resemblance to the work of B. Marchena and C. Piñeiro (Quaestiones Math., to appear), where the results originated from the authors' interest in sequences in the ranges of vector measures.
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.