The existence of an orthonormal sequence.

Question

Let $H$ be a Hilbert space and assume that $T\in L(H)$ is a normal operator. Suppose that $\lambda \in \sigma(T)$ is an accumulation point in the sense that for any $\epsilon > 0$, the set $(\lambda - \epsilon,\lambda + \epsilon) \cap \sigma(T)$ is infinite. How do we prove that there is an orthonormal sequence $(x_n)_{n=1}^{\infty}$ in $H$ such that $\lim_{n\rightarrow \infty} \|(T- \lambda)x_n\|=0$ ?

Answer 1

What you need is the Borel functional calculus, and this answer assumes you have seen it. Your assumption implies that there is a sequence of positive real numbers $(r_n)$ converging to $0$ such that each set $$ A_n := \{z\in \sigma(T) : r_n > |z-\lambda| > r_{n+1}\} $$ is non-empty. Let $P_n$ denote the projection associated to $A_n$ by the Borel functional calculus, then each $P_n$ is non-zero, the $P_n$ are orthogonal to each other, and $$ \|(T-\lambda)P_n\| \leq r_n $$ Now simply choose unit vectors $x_n \in P_n(H)$, and they will do the job.