Transfinite ranges and the local spectrum (Q691847)

Let \(T\) be a bounded linear operator acting on a complex Banach space \(X\). For ordinal numbers \(\alpha\), the authors define the \(\alpha\)-ranges \(R^\alpha(T)\) which generalize the ranges of powers \(R(T^n)\). The coeur algébrique of \(T\) is defined by \(co(T)=\bigcap_\alpha R^\alpha(T)=R^{dsc(T)}(T)\), where the descent \(dsc(T)\) is the smallest ordinal number \(\alpha\) for which \(R^{\alpha+1}(T)=R^\alpha(T)\). The authors investigate the coeur algébrique, which has similar properties as the coeur analytique. They also introduce algebraic local spectra which have properties analogous to those of the classical analytic local spectra studied in local spectral theory.