On transitive operator algebras in real Banach spaces

TL;DR

This paper classifies weakly closed transitive operator algebras with compact operators in real Banach spaces into real, complex, and quaternion types, providing properties, characterizations, and examples, including a continuum of non-similar algebras in separable real Hilbert spaces.

Contribution

It introduces a classification of Lomonosov algebras in real Banach spaces and explores their properties and diversity, especially in separable real Hilbert spaces.

Findings

Algebras are divided into real, complex, and quaternion classes.

In separable real Hilbert spaces, there are continuum many non-similar complex and quaternion type algebras.

Abstract

We consider weakly closed transitive algebras of operators containing non-zero compact operators in real Banach spaces (Lomonosov algebras). It is shown that they are naturally divided in three classes: the algebras of real, complex and quaternion classes. The properties and characterizations of algebras in each class as well as some useful examples are presented. It is shown that in separable real Hilbert spaces there is a continuum of pairwise non-similar Lomonosov algebras of complex type and of quaternion type.