Transfinite ranges and the local spectrum

Let T be a Banach space operator. For ordinal numbers α we define the α-ranges Rα(T) which generalize the ranges of powers R(Tn). The intersection ⋂αRα(T) is the coeur algébrique of T. Moreover, the coeur algébrique (and more generally the α-ranges for limit ordinals α) have similar properties as the coeur analytique of T. So it is possible to introduce algebraic local spectra which have properties analogous to those of classical (analytic) local spectra studied in local spectral theory.