Grothendieck’s theorem for absolutely summing multilinear operators is optimal

Abstract

Grothendieck’s theorem asserts that every continuous linear operator from to is absolutely -summing. In this note, we prove that the optimal constant so that every continuous -linear operator from to is absolutely -summing is . We also show that if there is dimensional linear space composed by continuous non absolutely -summing -linear operators from to In particular, our result solves (in the positive) a conjecture posed by A.T. Bernardino in 2011.

Keywords:

• lineability
• spaceability
• absolutely summing operators

Acknowledgements

The authors thank Prof. G. Botelho and the anonymous referee for important suggestions.

Notes

D. Pellegrino and J.B. Seoane-Sepúlveda were supported by CNPq [grant number 401735/2013-3] (PVE- Linha 2).