Limiting absorption principle for perturbed operator

Abstract

In this note the following theorem is proved. Let $\mathcal{H}$ and $\mathcal{K}$ be Hilbert spaces. Let $H_0$ be a self-adjoint operator on $\mathcal{H},$ $F:\mathcal{H}\to \mathcal{K}$ be a closed $|H_0{|}^{1/2}$-compact operator, and $J:\mathcal{K}\to \mathcal{K}$ be a bounded self-adjoint operator. If the operator $$F(H_0-\lambda -iy{)}^{-1}{F}^{\ast }$$ has norm limit as $y\to 0^{+}$ for a.e.~$\lambda ,$ then so does the operator $$F(H_0+F^{\ast }JF-\lambda -iy{)}^{-1}{F}^{\ast }.$$ An invariant operator ideal version of this result is also discussed.

4 pages; an invariant operator ideal version of the main result has been added

References in corpus (1)

• Spectral flow inside essential spectrum III: coupling resonances near essential spectrum