Weyl-von Neumann Theorem and Borel complexity of unitary equivalence modulo compact perturbations of self-adjoint operators
Speaker: Hiroshi Ando, University of Copenhagen.
The celebrated Weyl-von Neumann perturbation theorem asserts that every self-adjoint operator can be turned into a diagonal operator by an arbitrarily small compact (in fact any Schatten p-class, p>1) perturbations. As a corollary, von Neumann obtained the following result: for bounded self-adjoint operator A,B, the following conditions are equivalent: (1) uAu^*+K=B, where u is a unitary and K is a compact operator. (2) A and B have the same essential spectrum. Here, the essential spectrum of A is the set of all points in the spectrum of A which is either an eigenvalue of infinite multiplicity or an accumulation point in the spectrum. Therefore, the essential spectrum is a complete invariant for the classification problem of bounded self-adjoint operators in the sense of (1).
Since Weyl-von Neumann theorem works for general unbounded self-adjoint operators, it is interesting to consider whether von Neumann's theorem still holds for unbounded operators. However, many examples indicate us that there is very little hope to find an invariant I(A) (assigned in a constructible way) for each self-adjoint A such that for A,B uAu*+K=B holds for some u,K iff I(A)=I(B).
The purpose of our work is to justify this feeling by showing: (a) It can be deduced from von Neumann's characterization that the classification of bounded self-adjoint operators up to unitary equivalence modulo compacts is smooth. (b) The same type of equivalence relation for unbounded self-adjoint operators does not admit classification by countable structure, although the associated orbit equivalence relation is not generically turbulent.
This is a joint work with Yasumichi Matsuzawa (Shinshu University).
Since Weyl-von Neumann theorem works for general unbounded self-adjoint operators, it is interesting to consider whether von Neumann's theorem still holds for unbounded operators. However, many examples indicate us that there is very little hope to find an invariant I(A) (assigned in a constructible way) for each self-adjoint A such that for A,B uAu*+K=B holds for some u,K iff I(A)=I(B).
The purpose of our work is to justify this feeling by showing: (a) It can be deduced from von Neumann's characterization that the classification of bounded self-adjoint operators up to unitary equivalence modulo compacts is smooth. (b) The same type of equivalence relation for unbounded self-adjoint operators does not admit classification by countable structure, although the associated orbit equivalence relation is not generically turbulent.
This is a joint work with Yasumichi Matsuzawa (Shinshu University).