On Burnside type theorems in real Banach spaces. Lomonosov algebras

We consider weakly closed transitive algebras of operators in real Banach spaces containing non-zero compact operators (Lomonosov algebras). It will be shown that they are naturally divided in three classes: the algebras of real, complex and quaternion types. The properties and characterizations of algebras in each class as well as some useful examples are presented. It is shown that in any Banach space there is only one Lomonosov algebra of real type and that in separable real Hilbert spaces there is a continuum of pairwise non-similar Lomonosov algebras of complex type and of quaternion type.
(A joint work with Edward Kissin and Yurii Turovskii)