Polynomial sequences of bounded operators

The basic results of spectral theory are obtained using the sequence of powers of a bounded linear operator T,T2,…,Tn,…. In this paper, we replace the powers Tn by certain polynomials pn(T), and make use of special properties of the polynomial sequence { pn}n⩾0 to derive some new results concerning operators. For example, using an arbitrary polynomial sequence { pn}n⩾0, we obtain “binomial” spectral radii and semidistances, which reduce, in the case of the sequence of powers, to the usual spectral radius and semidistance.