Jan H. Fourie

Banach space sequences and projective tensor products

Abstract In this paper we use results from the theory of tensor products of Banach spaces to establish the isometry of the space of (1,p)-summing sequences (also known as strongly p-summable sequences) in a Banach space X, the space of nuclear X-valued operators on lp and the complete projective tensor product of lp with X. Through similar techniques from the theory of tensor products, the isometry of the sequence space Lp〈X〉 (recently introduced in a paper by Bu, Quaestiones Math. (2002), to appear), the space of nuclear X-valued operators on Lp(0,1) (with a suitable equivalent norm) and the complete projective tensor product of Lp(0,1) with X is established. Moreover, we find conditions for the space of (p,q)-summing multipliers to have the GAK-property (generalized AK-property), use multiplier sequences to characterize Banach space valued bounded linear operators on the vector sequence space of absolutely p-summable sequences in a Banach space and present short proofs for results on p-summing multipliers.

Tensor Products and Banach Ideals of p-Compact Operators.

AbstractA study is made of the norm wp (1 ≤ p ≤ ∞) on the tensor product of two Banach spaces E and F. It is shown that wp is a tensor norm, and a representation is deduced for the elements in the completion

DP*-properties of order p on Banach spaces

E\tilde \otimes _{w_p } F

Projections and embeddings of locally convex operator spaces and their duals

of E ⊗ F equipped with wp. Finally it is shown that the wp-nuclear operators in the sense of Grothendieck [3] coincide with those operators factoring compactly throughp (if 1 ≤ p ≤ ∞) or Co (if p=∞), with related equalities concerning the idea1 norms.

On weak-star p -convergent operators

An elementary proof of the (known) fact that each element of the Banach spaceℓwp(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 ofℓwp(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.

PROJECTIONS ON OPERATOR DUALS

Abstract In this article a new property of Banach spaces, called the DP*-property of order p (briefly denoted by DP* Pp) is introduced and characterisations of Banach spaces with this property and some applications thereof to polynomials and holomorphic functions on Banach spaces are studied.

ABSOLUTELY SUMMING MULTIPLIERS AND TOPOLOGIES ON L(E,F)

ABSTRACT Let Λ be a scalar sequence space which is endowed with a normal locally convex topology. For a separated locally convex space E we denote by Λ(E) the vector space of all sequences g in E for which (>g(i),a x, f(i)<) e Λ; for all x e E, when E is barrelled. We conclude the paper by application of the results on vector sequence spaces to spaces of operators—including for instance, necessary and...

Classes of sequentially limited operators

Let E , F be Hausdorff locally convex spaces. In this note we consider conditions on E and F such that the dual space of the space K b ( E , F ) (of quasi-compact operators) is a complemented subspace of the dual space of L b ( E , F ) (of continuous linear operators). We obtain necessary and sufficient conditions for L b ( E , F ) to be semi-reflexive.