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I am looking for an example of an (infinite) matrix that has a residual spectrum. For context, I know that the diagonal matrix $A$ with $A_{ii} = \frac{1}{i}$ has a point spectrum consisting of $\frac{1}{i}$ for $i \in \mathbb{N}$, and 0 is part of the continuous spectrum.

Given this, I am wondering if there is a simple example of a matrix that also has a residual spectrum. I understand that the matrix cannot be self-adjoint, but I am unsure of how to construct or find such an example.

Any help would be greatly appreciated!

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    $\begingroup$ Notational comment: When taking about spectral theory, and hence about complex numbers, using $i$ is an index is not advisable. $\endgroup$ Commented Mar 1 at 18:59
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    $\begingroup$ Why is there a "random matrices" tag? $\endgroup$ Commented Mar 1 at 21:14

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If you consider the right shift $V$ acting on $\ell_2$ then $\sigma_r(V)=\{z \in \mathbb{C} : |z|<1\}$. Note that $V$ is a linear isometry with closed proper range.

The associated matrix: $a_{k+1,k}=1$ for each $k\in\mathbb{N}$ and $a_{k,l}=0$ otherwise.

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