Limiting absorption principle for perturbed operator

In this note the following theorem is proved. Let $\mathcal{H}$ and $\mathcal K$beHilbertspaces\mathrm{.}Let$H_0$beaself-adjointoperatoron$\mathcal H,$ $F \colon \mathcal H \to \mathcal K$beaclosed$|H_0|^{1/2}$-compact operator, and $J\text{\hspace{0.17em}⁣}:\mathcal{K}\to \mathcal{K}$ 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 $$F(H_0+F^{\ast }JF-\lambda -iy{)}^{-1}{F}^{\ast }\mathrm{.}$$ An invariant operator ideal version of this result is also discussed.