Principal part of resolvent and factorization of an increasing nonanalytic operator-function |
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Authors: | Alexander Markus Vladimir Matsaev |
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Affiliation: | (1) Department of Mathematics and Computer Sciences, Ben-Gurion University of the Negev, Beer-Sheva, Israel;(2) Raymond and Beverly Sackler Faculty of Exact Sciences School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv, Israel |
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Abstract: | Let a selfadjoint operator-valued functionL() be given on the interval [a,b] such thatL(a)0,L(b)0,L()0 (ab), andL() has a certain smoothness (for instance, it satisfies Hölder's condition). It turns out that the spectral theory of the operator-valued functionL() can be reduced to the spectral theory of one operatorZ, the spectrum of which lies on (a, b) and which is similar to a selfadjoint operator. In particular, the factorization takes place:L()=M()(I–Z), where the operator-valued functionM() is invertible on [a, b]. Earlier similar results were known only for analytic operator-valued functions. The authors had to use new methods for the proof of the described theorem. The key moment is the decomposition ofL–1() into the sume of its principal and regular parts. |
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