# Constructing an approximation of an eigenvector

There's a theorem I should prepare for the oral exam, however I cannot find it anywhere in the recommended literature.

I should prove the following:

Let X be a complex Hilbert space and $$A \in B(X), \lambda \in \sigma_c(A)$$. Then, there exists a sequence $$(x_n)_{n \in \mathbb{N}}$$ in X such that $$\lim_{n \to \infty}(\lambda I - A) x_n = 0$$ and $$|| x_n|| =1$$ for every $$n \in \mathbb{N}$$.

I'd approach this by constructing such a sequence, but I'm not sure how to.

That's the definition of $$\sigma_c(A)$$. $$\lambda$$ is in the continuous spectrum if $$\mathcal{N}(\lambda I-A)=\{0\}$$, but the inverse operator is not bounded. Because the inverse is not bounded, there is a sequence $$\{ x_n \}$$ of unit vectors in the domain of $$\lambda I-A$$ such that $$(\lambda I-A)x_n\rightarrow 0$$.
Example: Let $$X=L^2[0,1]$$ and let $$A : X\rightarrow X$$ be the multiplication operator $$A : X \rightarrow X$$ defined by $$(Af)(x) = xf(x)$$. Then the spectrum of $$A$$ is $$[0,1]$$, and it is entirely continuous spectrum. If you choose $$\lambda \in (0,1)$$, then you can see that you nearly have an eigenvector $$x_{\lambda,\epsilon}$$ with eigenvalue $$\lambda$$ defined by $$x_{\lambda,\epsilon}=\chi_{[\lambda-\epsilon,\lambda+\epsilon]}.$$ This is because $$\|(A-\lambda I)x_{\lambda,\epsilon}\|^2 = \int_{0}^{1}(x-\lambda)^2\chi_{[\lambda-\epsilon,\lambda+\epsilon]}(x)^2dx \\ \le \epsilon^2\int_0^1\chi_{[\lambda-\epsilon,\lambda+\epsilon]}(x)^2 dx = \epsilon^2\|\chi_{\lambda,\epsilon}\|^2.$$ That is, $$\|(A-\lambda I)x_{\lambda,\epsilon}\| \le \epsilon\|x_{\lambda,\epsilon}\|.$$ So every $$\lambda\in[0,1]$$ is an approximate eigenvalue, but not an actual one. I'll let you consider the case of other Lebesgue measures, not just absolutely continuous ones.
• @NikolaBurbakić : I made a correction to my post. I wrote that the unit vectors were in the range. I meant that they were in the domain, so that $(\lambda I-A)x_n$ makes sense. This is sometimes referred to as having an approximate eigenvalue $\lambda$. Aug 5 at 22:03