Resolvent estimate of compact perturbation of self-adjoint operator

Question

Let $T$ be a selfadjoint operator on Hilbert space $H$. Then we know that there is a resolvent estimation $$\left\lVert (\lambda-T)^{-1}\right\rVert = \frac{1}{dist(\lambda,\sigma(T))}, \ \lambda \in \rho(T).$$ But what if we want to consider the compact perturbation, that is, operator $A = T+ D,$
where $D$ is a compact operator on $H$. Is there any known results on the resolvent estimate? Does anyone know some related references ? Thank you very much.