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Summing multi-norms defined by Orlicz spaces and symmetric sequence space

Blasco O.

Commentationes Mathematicae

2016

Vol 56, No. 1

145--168

We develop the notion of the (X1,X2)-summing power-norm based on a~Banach space E, where X1 and X2 are symmetric sequence spaces. We study the particular case when X1 and X2 are Orlicz spaces ℓΦ and ℓΨ respectively and analyze under which conditions the (Φ,Ψ)-summing power-norm becomes a~multinorm. In the case when E is also a~symmetric sequence space L, we compute the precise value of ∥(δ1,⋯,δn)∥n(X1,X2), where (δk) stands for the canonical basis of L, extending known results for the (p,q)-summing power-norm based on the space ℓr which corresponds to X1=ℓp, X2=ℓq, and E=ℓr.