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.


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. $\lambda \in \rho(U)$.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.