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Analytic joint spectral radius in a solvable Lie algebra of operators
by
Daniel Beltita
Institute of Mathematics of the Romanian Academy
The algebraic joint spectral radius of a set of operators (introduced by G.-C. Rota and G. Strang) has proved recently its usefulness in the solution of an important open problem concerning the existence of invariant subspaces for semigroups of compact quasinilpotent operators, cf. the paper of Yu.V. Turovskii in J. Funct. Anal. 162(1999), 313-322. On the other hand, several mathematicians have studied a geometric joint spectral radius for commuting tuples of operators (defined by means of the Taylor spectrum) and compared it with the algebraic one. Cf. the papers of R. Bhatia, T. Bhattacharyya, E. Boasso, M. Cho, V. Müller, A. Soltysiak, P. Rosenthal, W. Zelazko. Our aim here is to introduce the concept of analytic spectral radius for a family of operators indexed by some finite measure space. This spectral radius is compared with the algebraic and geometric ones when the operators belong to some finite-dimensional solvable Lie algebra (extending the framework of commutativity by means of the recently introduced Cartan-Taylor joint spectrum). We describe several situations when the three spectral radii coincide. These results extend some of the corresponding facts concerning commuting n-tuples of operators.
Date received: March 30, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeo-12.