On the unitary equivalence of absolutely continuous parts of self-adjoint extensions |
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Authors: | Mark M Malamud |
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Institution: | a Institute of Applied Mathematics and Mechanics, Universitetskaya str. 74, 83114 Donetsk, Ukraine b Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, D-10117 Berlin, Germany |
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Abstract: | The classical Weyl-von Neumann theorem states that for any self-adjoint operator A0 in a separable Hilbert space H there exists a (non-unique) Hilbert-Schmidt operator C=C? such that the perturbed operator A0+C has purely point spectrum. We are interesting whether this result remains valid for non-additive perturbations by considering the set ExtA of self-adjoint extensions of a given densely defined symmetric operator A in H and some fixed . We show that the ac-parts and of and A0 are unitarily equivalent provided that the resolvent difference is compact and the Weyl function M(⋅) of the pair {A,A0} admits weak boundary limits M(t):=w-limy→+0M(t+iy) for a.e. t∈R. This result generalizes the classical Kato-Rosenblum theorem. Moreover, it demonstrates that for such pairs {A,A0} the Weyl-von Neumann theorem is in general not true in the class ExtA. |
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Keywords: | Symmetric operators Self-adjoint extensions Boundary triplets Weyl functions Unitary equivalence |
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