Journal of Operator Theory

Volume 92, Issue 2, Autumn 2024 pp. 305-329.

On transitive operator algebras in real Banach spaces

Authors: Edward Kissin (1), Victor S. Shulman (2), and Yurii V. Turovskii (3)
Author institution: (1) London Metropolitan University, 166-220 Holloway Road, London N7 8DB, U.K.
(2) Department of Mathematics, Vologda State University, Vologda, Russia
(3) Department of Mathematics, Vologda State University, Vologda, Russia

Summary: 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.

DOI: http://dx.doi.org/10.7900/jot.2021dec12.2475
Keywords: real Banach space, spectrum, operator algebra, invariant subspace, density theorem

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