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

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    $\begingroup$ 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. $\endgroup$ Commented Apr 14, 2012 at 13:44

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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 $\partial \sigma(T) = \sigma(T) \supseteq \sigma_{\text{ap}}(T)$.

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