A note on the relationship between spectral radius and norms of bounded linear operators

Abstract

Let X be a Banach space and L ( X ) be the Banach algebra of bounded operators on X . In this note we prove that if we have a compact subset K of a commutative sub-algebra of L ( X ), and given " > 0, then it is possible to de ne a new norm in X , equivalent to its given norm, in such a way that inside a neighborhood U " of this compact set in the sub- algebra, the norms of all the operators di er from their spectral radius in less than " . If X is a Hilbert space then it is possible to de ne this new norm as an Hilbertian norm.

Postprint (published version)