Let $(H, \langle \cdot, \cdot \rangle)$ be a real Hilbert space and $|\cdot|$ its induced norm. Let $I:H \to H$ be the indentity map. Let $T:H \to H$ be a bounded linear operator. In preparation for Brezis' exercise 6.22.4, I would like to prove that
If $T$ is self-adjoint, then $T^2 + tI$ is bijective for any $t>0$.
There are possibly subtle mistakes that I could not recognize in below attempt. Could you please have a check on it?
First, we prove that $K:=T^2$ is positive, i.e., $\langle Ku, u \rangle \ge 0$ for all $u \in H$. Indeed, $$ \begin{align*} \langle Ku, u \rangle &= \langle T^2u, u \rangle \\ &= \langle Tu, Tu \rangle \text{ because } T \text{ is self-adjoint} \\ &= |Tu|^2 \ge 0. \end{align*} $$
Second, we verify that $K+tI$ is injective. Let $u\in H$ such $(K+tI)u=0$. Then $Ku = -t u$. Then $u=0$ because $$ 0 \le \langle Ku, u \rangle = -t \langle u, u \rangle = -t|u|^2. $$
Finally, we prove that $K+tI$ is surjective. We have $K^* = (T^2)^*=(T^*)^2=T^2=K$, so $K$ is self-adjoint. We have $(K+tI)^*= K^*+t I= K+tI$, so $K+tI$ is self-adjoint. For $u\in H$, we have $$ \begin{align*} |(K+tI)u|^2 &= |Ku|^2 +2t \langle Ku, u \rangle + t^2|u|^2 \\ &\ge t^2|u|^2 + 2t \langle Ku, u \rangle \\ &\ge t^2|u|^2 \text{ because } K \text{ is positive}. \end{align*} $$
The claim then follows from below Theorem 2.21. in the same book, i.e.,
Let $E, F$ be real Banach spaces. For a linear map $T$, we denote by $N(T)$ its kernel and by $R(T)$ its range. Let $A: D(A) \subset E \to F$ be an unbounded linear operator that is densely defined and has a closed graph.
Theorem 2.20. The following properties are equivalent:
- (a) $A$ is surjective, i.e., $R(A)=F$,
- (b) there is a constant $C$ such that $$ |v| \leq C |A^* v| \quad \forall v \in D (A^*), $$
- (c) $N (A^*) = \{0\}$ and $R(A^*)$ is closed.
Theorem 2.21. The following properties are equivalent:
- (a) $A^*$ is surjective, i.e., $R(A^*)=E^*$,
- (b) there is a constant $C$ such that $$ |u| \leq C |A u| \quad \forall u \in D (A), $$
- (c) $N (A) = \{0\}$ and $R(A)$ is closed.