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