When reading the proof the following Corollary, I met with some problems. Notation : $\mathfrak{S}$ denotes the class of operators of the form $\lambda I+K$, where $0\neq \lambda \in \Bbb C$ and $K$ is a compact operator. It is easy to check that $W_e(T)=\{\lambda\}$ iff $W_e(T-\lambda I)=\{0\}$, but how to use the lemma to prove that $W_e(T-\lambda I)=\{0\}$ iff $T\in \mathfrak{S}$.
In order to show that $T\in \mathfrak{S}$, we need to verify that $T-\lambda I$ is compact, that is to say, for every orthonomal set $\{e_n\}$, we have $((T-\lambda I)e_n, e_n)\rightarrow 0$. As $W_e(T-\lambda I)=\{0\}$, according to the definition of essential numerical range of $T-\lambda I$, there exists an orthonormal set $\{f_n\}$ such that$((T-\lambda I)f_n, f_n)\rightarrow 0$.
My question: how to show that $(T-\lambda I)e_n, e_n)\rightarrow 0$for every orthonomal set $\{e_n\}$ ?
Suppose that there exists an orthonormal set $\{e_n\}$ such that $\langle (T-\lambda I)e_n,e_n\rangle\not\to0$. Since this sequence of numbers is bounded, it has a convergent subsequence that converges to some $\mu\ne0$. Then $\mu\in W_e(T-\lambda I)$, which means that $\lambda+\mu\in W_e(T)$, contradicting the hypothesis.