A note on the relationship between spectral radius and norms of bounded linear operators
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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.
CitationMunhoz, H.; Sola-morales, J. A note on the relationship between spectral radius and norms of bounded linear operators. "Matemática contemporânea", 2009, vol. 36, p. 131-137.