Vol. 25, No. 1, 1968

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Sum and product of commuting spectral operators

Kirti K. Oberai

Vol. 25 (1968), No. 1, 129–146

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.

Mathematical Subject Classification
Primary: 47.30
Received: 31 October 1966
Published: 1 April 1968
Kirti K. Oberai