I have a question concerning the approximate point 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 $\|(T-\lambda)u_n\|\to 0$ as $n\to \infty.$ We denote this set by $\sigma_{ap}(T)$. We denote the well-known spectrum of $T$ by $\sigma(T)$. It is know that $\sigma(T)$ is non empty and that $\sigma_{ap}(T)\subset\sigma(T)$
My question : Can we prove that $\sigma_{ap}(T)$ is non empty?
Any help is appreciated. Thanks in advance.
Best Answer
As mentioned by Ryszard Szwarc in the comments, the boundary of the spectrum is always contained in the approximate point spectrum, so the latter is non-empty. This result is Proposition VII.6.7 in Conway's book A Course in Functional Analysis.