Let $T$ be a selfadjoint operator on Hilbert space $H$. Then we know that there is a resolvent estimation $$\left\lVert (\lambdaT)^{1}\right\rVert = \frac{1}{dist(\lambda,\sigma(T))}, \ \lambda \in \rho(T).$$
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.
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1$\begingroup$ There are lots of standard estimates using variants of the neumann series for bounded $D$. Also for compact $D$ if $A$ has compact resolvent. See the monograph of Kato. springer.com/de/book/… $\endgroup$ – András Bátkai Jan 8 at 5:44

3$\begingroup$ Can you specify what you are looking for? $\endgroup$ – András Bátkai Jan 8 at 5:44

$\begingroup$ @AndrásBátkai Thank you very much for your comment. I am looking for a bound on $\left\lVert (\lambda  (T + D))^{1} \right\rVert$, where $T$ is selfadjoint operator and $D$ is a compact operator(we can add more conditions on the operators if it is necessary). And I hope the bound can involve $dist(\lambda,\sigma(T+D))$, like the one I write above. Or more precisely, I expect a result similar as the one in maths.qmul.ac.uk/~ob/publ/carleman.pdf $\endgroup$ – Bowen Jan 8 at 7:36