Denote $L$ as the (uniformly) elliptic operator. The following are two theorems in Evans's Partial Differential Equations:
Here $\Sigma$ is the (real) spectrum of the operator $L$.
According to the answer to a previous question of mine, in Theorem 3, the operator $L+\mu I$ gives a Banach space isomorphism between $H^1_0$ and $H^{-1}$ as long as $\mu>\gamma$ where $\gamma\geq 0$ is some constant depends on $L$.
In Theorem 6, as long as $\mu\not\in\Sigma$, one has on bounded operator $(L-\mu I)^{-1}$ on $L^2(U)$.
These two theorems look so similar. Are there any "deep" relation between them?
As long as $\mu > \gamma$, Theorem 6 follows from the answer to your previous question. Namely, you have an isomorphism $L - \mu I \colon H^1_0(U) \rightarrow H^{-1}(U)$ and you have a continuous embeddings $L^2(U) \hookrightarrow H^{-1}(U)$ and $H^1_0(U) \rightarrow L^2(U)$ and thus you get a bounded operator
$$ L^2(U) \hookrightarrow H^{-1}(U) \xrightarrow[]{(L - \mu I)^{-1}} H^1_0(U) \hookrightarrow L^2(U). $$