arXiv subjects
Publications and source records attributed to Yuri V. Turovskii.
The work starts a series of papers on topological radicals and their applications. Among other results we present a theory of radicals related to the joint tensor radius.
We develop the spectral radius technique and the theory of tensor radicals. As applications we obtain numerous results on mutiplication operators in Banach algebras and Operator bimodules.
We develop the theory of topological radicals in Banach Lie algebras, consider various ways of constructing such radicals and study a family of (pre)radicals related to subalgebras and ideals of finite codimension in Banach Lie algebras. It extends to the infinite-dimensional case many results of the well known Frattini theory of Lie algebras.
We introduce and study procedures and constructions in the theory of the joint spectral radius that are related to the spectral theory. In particular we devlop the theory of the scattered radical. Among applications we find some sufficient conditions of continuity of the spectrum and spectral radii of various types, and prove that in GCR C*-algebras the joint spectral radius is continuous on precompact subsets and coincides with the Berger-Wang radius.
It is shown that the joint spectral radius $ρ(M)$ of a precompact family $M$ of operators on a Banach space $X$ is equal to the maximum of two numbers: the joint spectral radius $ρ_{e}(M)$ of the image of $M$ in the Calkin algebra and the Berger-Wang radius $r(M)$ defined by the formula \[ r(M)=\underset{n\to\infty}{\limsup}(\sup\left\{ρ(a):a\in M^{n}\right\} ^{1/n}) . \] Some more general Banach-algebraic results of this kind are also proved. The proofs are based on the study of special radicals on the class of Banach algebras.