On spectral radius algebras

Abstract

We show how one can associate a Hermitian operator P to every operator A , and we prove that the invertibility properties of P imply the non-transitivity and density of the spectral radius algebra associated to A . In the finite dimensional case we give a complete characterization of these algebras in terms of P . In addition, we show that in the finite dimensional case, the spectral radius algebra always properly contains the commutant of A .