𝖫^{2}-summand vectors in Banach spaces

Abstract

The aim of this paper is to study the set L X 2 \mathsf {L} _{X}^{\mathtt {2}} of all L 2 \mathsf {L}^{\mathtt {2}} -summand vectors of a real Banach space X X . We provide a characterization of L 2 \mathsf {L}^{\mathtt { 2}} -summand vectors in smooth real Banach spaces and a general decomposition theorem which shows that every real Banach space can be decomposed as an L 2 \mathsf {L}^{\mathtt {2}} -sum of a Hilbert space and a Banach space without nontrivial L 2 \mathsf {L}^{\mathtt {2}} -summand vectors. As a consequence, we generalize some results and we obtain intrinsic characterizations of real Hilbert spaces.