# Approximate point spectrum

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 $$\sigma_\text{ap}(T)\subset\partial\sigma(T)$$?

• So you are asking if it's possible to have $\sigma_{ap}(T) = \partial \sigma(T)$? This isn't always true (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. Commented Apr 14, 2012 at 13:44

You can have that $$\sigma_{\text{ap}}(T) \subseteq \partial \sigma(T)$$ when $$T$$ is self-adjoint since in this case $$\sigma(T) \subseteq \mathbb{R}$$, so $$\partial \sigma(T) = \sigma(T) \supseteq \sigma_{\text{ap}}(T)$$.