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In Brezis' Functional analysis, Sobolev spaces and partial differential eqautions, exercise 6.24 (3) asks to prove that for a self-adjoint operator $T\in \mathcal{L}(H)$, $H$ a Hilbert space, the following properties are equivalent

(vii) $(Tu, u)\leq |Tu|^2, u\in H$

(viii) $(0,1)\subset \rho(T)$

where $\rho(T)$ denotes the resolvent set of $T$. The book gives the following solution

Set $U=2T-I$. Clearly (vii) is equivalent to $$\text{(vii')} \;\;\;\;\;\;\;\;\;|u|\leq |Uu| \; \;\;\;\;\;\forall u\in H$$ Applying Theorem 2.20 we see that (vii)$\Rightarrow (-1,+1)\subset \rho(U)=2\rho(T)-1$. Thus (vii)$\Rightarrow$ (viii).

and then it proceeds to prove the converse, which I can understand perfectly. The part I don't get is how applying Theorem 2.20 gives $(-1,+1)\subset \rho (U)$ (everything else is clear). Theorem 2.20 states

Theorem 2.20: Let $A:D(A)\subset E\rightarrow F$ be an unbounded linear operator that is densely defined and closed. The following properties are equivalent:

(a) $A$ is surjective, i.e. $R(A)=F$,

(b) there is a constant $C$ such that $$\|v\|\leq C\|A^{\ast}v\| \;\; \forall v\in D(A^{\ast}),$$ (c) $N(A^{\ast})=\{0\}$ and $R(A^{\ast})$ is closed.

If someone could explain to me how Theorem 2.20 is applied or even give another solution for the (vii)$\Rightarrow$ (viii) part, I would be grateful.

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Remember that $T$, and hence $U = 2T-I$, is self-adjoint by assumption.

Now, for $\lambda \in (-1,1)$, by using (vii'), we have

$$\lVert (\lambda I - U)v\rVert \geqslant \lVert Uv\rVert - \lambda \lVert v\rVert \geqslant (1-\lvert\lambda\rvert)\cdot \lVert v\rVert \tag{$\ast$}$$

for all $v\in H = D(U)$. But $(\ast)$ is just condition b) of theorem 2.20, and the equivalence of that with conditions a) and c) shows that $\lambda I - U$ is invertible, i.e. $\lambda \in \rho(U)$.

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  • $\begingroup$ The first inequality is just the reverse triangle inequality along with the fact that $\|Uv\|-\lambda \|v\|\geq 0$ for $\lambda\in (-1,1)$, right? $\endgroup$ – John Doe Dec 11 '14 at 15:58
  • $\begingroup$ Reverse triangle inequality, then (vii'), which says $\lVert Uv\rVert \geqslant \lVert v\rVert$. We need a $c > 0$, namely $c = 1-\lvert\lambda\rvert$, to apply theorem 2.20. $\endgroup$ – Daniel Fischer Dec 11 '14 at 16:05
  • $\begingroup$ Got it, thanks! $\endgroup$ – John Doe Dec 11 '14 at 16:08

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