AbstractAbstr GRAPH SUBSPACES AND THE SPECTRAL SHIFT FUNCTION
نویسندگان
چکیده
We extend the concept of Lifshits–Krein spectral shift function associated with a pair of self-adjoint operators to the case of pairs of (admissible) operators that are similar to self-adjoint operators. An operator H is called admissible if: (i) there is a bounded operator V with a bounded inverse such that H = V −1 HV for some self-adjoint operator H; (ii) the operators H and H are resolvent comparable, i. e., the difference of the resolvents of H and H is a trace class operator (for non-real values of the spectral parameter); (iii) tr(V R − RV) = 0 whenever R is bounded and the commutator V R − RV is a trace class operator. The spectral shift function ξ(λ, H, A) associated with the pair of resolvent comparable admissible operators (H, A) is introduced then by the equality ξ(λ, H, A) = ξ(λ, H, A) where ξ(λ, H, A) denotes the Lifshits– Krein spectral shift function associated with the pair (H, A) of self-adjoint operators. Our main result is the following. Let H 0 and H 1 be separable Hilbert spaces, A 0 a self-adjoint operator in H 0 , A 1 a self-adjoint operator in H 1 , and B ij a bounded operator from H j
منابع مشابه
The Local Trace Function of Shift Invariant Subspaces
We define the local trace function for subspaces of L (Rn) which are invariant under integer translation. Our trace function contains the dimension function and the spectral function defined in [BoRz] and completely characterizes the given translation invariant subspace. It has properties such as positivity, additivity, monotony and some form of continuity. It behaves nicely under dilations and...
متن کاملPerturbation of Spectra and Spectral Subspaces
We consider the problem of variation of spectral subspaces for linear self-adjoint operators under off-diagonal perturbations. We prove a number of new optimal results on the shift of the spectrum and obtain (sharp) estimates on the norm of the difference of two spectral projections.
متن کامل[ m at h . SP ] 2 3 Ju l 2 00 7 PERTURBATION OF SPECTRA AND SPECTRAL SUBSPACES ∗
We consider the problem of variation of spectral subspaces for linear self-adjoint operators with emphasis on the case of off-diagonal perturbations. We prove a number of new optimal results on the shift of the spectrum and obtain (sharp) estimates on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators ...
متن کاملInvariant Manifolds Associated to Invariant Subspaces without Invariant Complements: a Graph Transform Approach
We use the graph transform method to prove existence of invariant manifolds near fixed points of maps tangent to invariant subspaces of the linearization. In contrast to the best known of such theorems, we do not assume that the corresponding space for the linear map is a spectral subspace. Indeed, we allow that there is no invariant complement (in particular, we do not need that the decomposit...
متن کامل