Hilbert space structure and positive operators

TL;DR

This paper explores the conditions under which a real Banach space can be characterized as Hilbertian based on the existence of certain positive operators, providing new insights into the structure of Banach spaces.

Contribution

It establishes a link between positive operators and the Hilbertian structure of Banach spaces, including non-symmetric cases, advancing understanding of space decompositions.

Findings

Existence of specific positive operators implies Hilbertian structure or subspace in Banach spaces.

Characterization of Banach spaces via positive operators and their symmetry properties.

Extension of results to non-symmetric operator cases.

Abstract

Let X be a real Banach space. We prove that the existence of an injective, positive, symmetric and not strictly singular operator from X into its dual implies that either X admits an equivalent Hilbertian norm or it contains a nontrivially complemented subspace which is isomorphic to a Hilbert space. We also treat the non-symmetric case.