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 ℓ p and the complete projective tensor product of ℓ p with X. Through similar techniques from the theory of tensor products, the isometry of the sequence space L p 〈 X〉 (recently introduced in a paper by Bu, Quaestiones Math. (2002), to appear), the space of nuclear X-valued operators on L p (0,1) (with a suitable equivalent norm) and the complete projective tensor product of L p (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.