Sum and product of commuting spectral operators

Kirti Oberai
1968 Pacific Journal of Mathematics  
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 \f\ p dμ) Dieudonne [1] obtained some of the properties of Ω 1 . By using his methods, we prove that the space Ω p ( = Ω P (X, μ))(l < p < ~) is a complete metrisable space and is also weakly sequentially complete. We also obtain the
more » ... al of Ω p . By using some inequalities obtained by McCarthy [7], we show that the sum and the product of two commuting scalar operators on Ω p (2 ^ p < oo) are again scalar opetators and the sum and the product of two commuting spectral operators are spectral operators provided that the spectrum of each operator is compact. 129 130 KIRTI K. OBERAI
doi:10.2140/pjm.1968.25.129 fatcat:x6qho6iz4re45dntnlxjad2y5i