Linear bijective maps preserving spectral functions on pairs of similar operators

Abstract

Let X be an infinite-dimensional complex Banach space, and let B(X) be the algebra of all linear and bounded operators on X. In this paper, we characterize linear bijective maps φ on B(X) having the property that the spectral radius (respectively, the spectrum) of φ(A) equals the spectral radius (respectively, the spectrum) of φ(B) for each pair of similar operators A,B∈B(X). As a corollary, we obtain a characterization of linear bijective maps on B(X) preserving the equality of the spectrum.