Sum and product of commuting spectral operators
Kirti K. Oberai · Pacific Journal of Mathematics · 1968
Let E be a separated, quasi-complete and barreled locally convex space.Let TΊ and T 2 be two commuting, continuous spectral operators on E. The conditions under which TΊ + T 2 and TιT 2 are spectral operators are obtained.Further, let X be a locally compact and <x-compact space.Let μ be a positive Radon measure on X.Let Ω*{X, μ)(l ^ p < oo) be the linear space of all complex valued functions defined on X, whose p th powers are locally integrable with respect to the measure μ.This space is given a certain topology under which it becomes a complete metrisable locally convex space.The sum and product of two commuting scalar operators on Ω P (X, μ)(2 ^ p < oo) are scalar operators and the sum and the product of two commuting spectral operators are spectral operators provided that the spectrum of each operator is compact.Proof.This is proved in [8, Lemma 2.15], An operator T e ^f(E) is said to commute with a spectral measure P( ) if P(σ)T = TP(σ) for all σe<^.