# approximate point spectrum

Good evening,

I have a question concerning the relation between 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 $\|(T-\lambda)u_n\|\to 0$ as $n\to \infty.$ We denote this set by $\sigma_{ap}(T)$. We denote the wellknown spectrum of $T$ by $\sigma(T)$.

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

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

Any help is appreciated. Thanks in advance.

Duc Anh

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What does "converse" mean. You could write (*) as $\sigma_{ap}(T) \supseteq \partial \sigma(T)$. Are you asking if the other inclusion, "$\subseteq$", is true? (It's not). – Matthew Daws Apr 14 '12 at 8:14
you're right. I'm sorry for the unclear question. I've editted it as above. Do you have a counter-example for it? – Đức Anh Apr 14 '12 at 9:27
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. – Matthew Daws Apr 14 '12 at 13:44
Thank you very much – Đức Anh Apr 14 '12 at 16:00