On the spectra of semi-normal operators
Calvin R. Putnam · Transactions of the American Mathematical Society · 1965
In particular T is normal if C (or D) is 0.By an isolated part a of sp(T) is meant a subset of sp(T) which lies at a positive distance from its complementary part sp(T) -a; see pp. 418 ff.].It is known that if a is an isolated part of sp(T), then there exists a "parallel projection" P = Pa, a bounded operator, not necessarily self-adjoint, satisfying P2 -P and such that both P §> and (/ -P) § are invariant under T. Moreover sp(F') = a, where T' = T/P%) denotes the restriction of T to the space P §.In case T is semi-normal, so also is T; cf.Berberian [1, p. 161, problem 10].In case A is a self-adjoint operator with the spectral resolution (1.4) A=(xdE(X), then the set §" of elements / in § for which fl E(X)f ||2 is an absolutely continuous function of X is known to be a subspace of §>; see Halmos [2, p. 104].