First results on spectrally bounded operators

A linear mapping T from a subspace E of a Banach algebra into another Banach algebra is defined to be spectrally bounded if there is a constant M greater than or equal to 0 such that r(Tx) less than or equal to Mr(x) for all x is an element of E, where r((.)) denotes the spectral radius. We study some basic properties of this class of operators, which are sometimes analogous to, sometimes very different from, those of bounded operators between Banach spaces.