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Consider the operator $M$ on $\ell^2(\mathbb{Z})$ defined by for $u\in \ell^2(\mathbb Z)$ $$Mu(n)=\frac{1}{\vert n \vert+1}u(n).$$ This is a compact operator! Then, let $l$ be the left-shift and $r$ ...

Let $K$ be a compact self-adjoint operator on a Hilbert space $H$ such that for some normalized $x \in H$ and $\lambda \in \mathbb C:$ $\Vert Kx-\lambda x \Vert \le \varepsilon.$ It is well-known ...