Exercise about spectrum of selfadjoint operator.

Question

I'm stuck on an exercise about the spectrum of a selfadjoint operator on a Hilbert space.

The problem is the following:

Let $(X,\langle \cdot, \cdot\rangle)$ a Hilbert space and let $A \in B(H)$ a selfadjoint operator such that $\langle Ax,x \rangle \geq 0$ for every $x \in X$.

a) If $m=\inf_{\|x\|=1}\langle Ax,x \rangle$, prove that $\lambda<m \Rightarrow \lambda \in \rho(A)$.

b) If $M=\sup_{\|x\|=1}\langle Ax,x \rangle$, show that $\sigma(A) \subseteq [m,M]$

any help is appreciated. Thanks

Answer 1

Let $N=\{ (Ax,x) : \|x\|=1\}$. Suppose $\lambda \notin N^c$ (closure of $N$) so that there exists $\delta$ such that the following holds whenever $\|x\|=1$: $$ 0 < \delta \le |(Ax,x)-\lambda|=|((A-\lambda I)x,x)|\le \|(A-\lambda I)x\|\|x\| = \|(A-\lambda I)x\| $$ Then $\|(A-\lambda I)x\| \ge \delta \|x\|$ for all $x$. (The same holds for $\overline{\lambda}$ as well.) Use this to argue that $$ \{ \lambda \in\mathbb{C} : \mbox{dist}(\lambda,N) > 0\} \subseteq\rho(A) \implies \sigma(A)\subseteq N^{c}. $$