Let E be a separated,
quasi-complete and barreled locally convex space. Let T1 and T2 be two commuting,
continuous spectral operators on E. The conditions under which T1+ T2 and T1T2
are spectral operators are obtained. Further, let X be a locally compact and
σ-compact space. Let μ be a positive Radon measure on X. Let Ωp(X,μ)(1 ≦ p < ∞)
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 Ωp(X,μ)(2 ≦ p < ∞) 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.
