Upper Bounds for Neumann–Schatten Norms

Abstract

Let H and H aux be Hilbert spaces, H a nonnegative self-adjoint operator in H,α,s>0 and J a bounded linear transformation from the Hilbert space D(H s/2) (equipped with the graph scalar product of H s/2) to H aux. It is shown that the operator J(H+α)−t belongs to the Neumann–Schatten class of order p=2+2(u−t)/(t−s/2) provided s/2<t<u,t−s/2<u−t and J(H+α)−u is Hilbert–Schmidt operator. An upper bound for the pth order Neumann–Schatten norm of J(H+α)−t is derived. If J is a closed operator from D(H 1/2) to H aux and D(J)⊃D(H) then there exists a unique self-adjoint operator H J in H such that D(H J )⊂D(J) and ( $$\left( {H^J f,g} \right) = \left( {H^{1/2} f,H^{1/2} g} \right) + \left( {Jf,Jg} \right)_{{\text{aux}}} ,{\text{ }}f \in D(H^J ),{\text{ }}g \in D(J)$$ . Conditions which are sufficient in order that the operator (H J +α)−1−(H+α)−1 is compact and conditions which are sufficient in order that the wave operators W ±(H J ,H) exist and are complete are derived. Instead of (Jf,Jg)aux also certain other perturbation terms, not by necessity nonnegative, are considered. The special case when H equals the operator (−Δ) r in L 2(R d ) for any strictly positive real number and H J equals (−Δ) r +μ for some suitably chosen measure μ is discussed in detail. In particular, new results on existence and completeness of the wave operators W ±(−Δ+μ,−Δ) are obtained.