Let $H^{+}\subset H\subset H^{-}$ be three embedded and dense Hilbert spaces and let $A$ denote a linear operator with domain $D(A)$. Let $\eta\in\mathbb{R}, \eta\neq 0$ and suppose that $$A -i\eta\in H.$$ Furthermore, for every $v\in H^{+}$, $v = Bu$ with $B$ operator and $u\in H$, suppose to have $$(A -i\eta) v = u.$$ If the operator $B$ is bounded from $H^{-}$ to $H^{+}$, why we can say that $(A-i\eta)^{-1}$ is bounded on $H$ to itself?
This is written on my notes, but I don't understand why it is true. Could someone please help me?
Thank you!
Hint: by construction for all $u\in H$, $$(A-i\eta)^{-1}u=Bu. $$ Take norms in $H$ then use the continuity of the embeddings and of $B$.