I have a question concerning the relation between the approximate point spectrum and the spectrum of an operator.

Let $T$ be a bounded linear operator of a complex Hilbert space $H$. The approximate point spectrum of $T$ is the set of all values $\lambda \in \mathbb{C}$ such that there exists a sequence of unit vectors $u_n\in H$ so that $\lVert(T-\lambda)u_n\rVert\to 0$ as $n\to \infty$. We denote this set by $\sigma_\text{ap}(T)$. We denote the wellknown spectrum of $T$ by $\sigma(T)$.

We know that $\sigma_\text{ap}(T)$ contains the topological boundary of $\sigma(T)$.

**My question :** Can we have $\sigma_\text{ap}(T)\subset\partial\sigma(T)$?

alwaystrue (left shift on $\ell^2$). But if $T$ is compact then $\sigma(T)$ is the point spectrum of $T$, a sequence converging to $0$ (or finite)-- so it is true if $T$ is compact. $\endgroup$