Algebras of Power Series of Elements of a Lie Algebra and the Slodkowski Spectra

Abstract

Topological algebras of (convergent) power series of elements of a Lie algebra are introduced and the existence of continuous homomorphisms of these algebras into an operator algebra is studied. For the Slodkowski spectra, the spectral mapping theorem $$\sigma _{\delta ,k} (f(a)) = f(\sigma _{\delta ,k} (a)),\sigma _{\pi ,k} (f(a)) = f(\sigma _{\pi ,k} (a))$$ is proved for generators a of a finite-dimensional nilpotent Lie algebra of bounded linear operators under the condition that a family f of elements of a power series algebra is finite-dimensional. Bibliography: 22 titles.