Operator Integrals, Spectral Shift, and Spectral Flow
نویسندگان
چکیده
منابع مشابه
Multiple Operator Integrals and Spectral Shift
For a large class of admissible functions f : R 7→ C, the operator derivatives dj dxj f(H0 + xV ), where H0 and V are self-adjoint operators on a separable Hilbert space H, exist and can be represented as multiple operator integrals [1, 14]. Let M be a semi-finite von Neumann algebra acting on H and τ a semi-finite normal faithful trace on M. For H0 = H ∗ 0 affiliated with M and V = V ∗ in the ...
متن کاملThe Spectral Shift Operator
We introduce the concept of a spectral shift operator and use it to derive Krein's spectral shift function for pairs of self-adjoint operators. Our principal tools are operator-valued Herglotz functions and their logarithms. Applications to Krein's trace formula and to the Birman-Solomyak spectral averaging formula are discussed.
متن کاملSome Applications of the Spectral Shift Operator
The recently introduced concept of a spectral shift operator is applied in several instances. Explicit applications include Krein's trace formula for pairs of self-adjoint operators, the Birman-Solomyak spectral averaging formula and its operator-valued extension, and an abstract approach to trace formulas based on perturbation theory and the theory of self-adjoint extensions of symmetric opera...
متن کاملSpectral Averaging and the Krein Spectral Shift
We provide a new proof of a theorem of Birman and Solomyak that if A(s) = A0+ sB with B ≥ 0 trace class and dμs(·) = Tr(BEA(s)(·)B), then ∫ 1 0 [dμs(λ)] ds = ξ(λ)dλ where ξ is the Krein spectral shift from A(0) to A(1). Our main point is that this is a simple consequence of the formula: d ds Tr(f(A(s)) = Tr(Bf ′(A(s))). Let A and C = A+B be bounded self-adjoint operators and suppose that B ≥ 0 ...
متن کاملEstimates for the spectral shift function of the polyharmonic operator
The Lifshits–Krein spectral shift function is considered for the pair of operators H0 = (−4)l, l > 0 and H = H0 + V in L2(R), d ≥ 1; here V is a multiplication operator. The estimates for this spectral shift function ξ(λ;H,H0) are obtained in terms of the spectral parameter λ > 0 and the integral norms of V . These estimates are in a good agreement with the ones predicted by the classical phase...
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ژورنال
عنوان ژورنال: Canadian Journal of Mathematics
سال: 2009
ISSN: 0008-414X,1496-4279
DOI: 10.4153/cjm-2009-012-0