Summary
We study the characteristic set
of a couple (A, B) of selfadjoint compact operators on a real Hilbert spaceH. We prove thatC is the union of a sequence of characteristic curvesC n in the (α, β) plane. Each curve is the analytic image of an open interval and it is either closed or it goes to infinity at both ends of the interval. Moreover, it may intersect either itself or other characteristic curves in an at most countable set of points, which may accumulate only at infinity. Finally, to each characteristic curve one can associate an analytic function En, which gives the eigenprojection onto the eigenspace attached to each point of the characteristic curve, except at the intersection points, where the eigenspace is the direct sum of the projection relevant to each branch passing through the point. The dimension of the eigenprojection is constant along each curve and it is called the multiplicity of the characteristic curve.
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Buzano, E. A two-parameter spectral theorem. Annali di Matematica 161, 139–151 (1992). https://doi.org/10.1007/BF01759635
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DOI: https://doi.org/10.1007/BF01759635