On the type of spectral operators and the non-spectrality of several differential operators on Lp

Abstract

A spectral operator, not necessarily bounded, which is the infinitesimal generator of a strongly continuous group of operators {U(t)|t∈ℝ} where ∥U(t)∥ = σ(|t|K), is of type k ∈ No, i.e. T = S + N, where S is spectral of scalar type, N is bounded, Nk+1 = O and S and N are commuting. This result yields a simple proof of the non-spectrality of certain differential operators on Lp, p ≠ 2, which are known to be selfadjoint for p = 2.