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On the theory of two-parameter spectral problems. (English. Russian original) Zbl 0626.47023

Sov. Math., Dokl. 32, 225-227 (1985); translation from Dokl. Nauk SSSR 283, 1033-1035 (1985).
Certain interesting problems of multiparameter spectral theory have been best studied in the particular two-parameter case. These include, for example, the investigations of Cordes on an abstract method for separation of variables, and the investigations of Faiermann on uniform convergence of eigenfunction expansions for two-parameter Sturm-Liouville equations. In the present note we give some results on the spectral theory of two-parameter problems. The main investigative tool involves passage from operator-valued coefficients of indefinite sign in general to special operators of constant sign, with preservation of the “weight” \(\Delta_ 0\).

MSC:

47A70 (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces
47A10 Spectrum, resolvent
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