Transfinite ranges and the local spectrum

Let T be a Banach space operator. For ordinal numbers α we define the α -ranges R α ( T ) which generalize the ranges of powers R ( T n ) . The intersection ⋂ α R α ( 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.