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.
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).