Isolated Eigenvalues on the Extensions.

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I asked this question on Mathoverflow https://mathoverflow.net/questions/226484/isolated-eigenvalue-of-t-is-also-an-isolated-eigenvalue-of-overlinet and because of the comments apparently the answer is trivial and the discussion does not belongs to there. But even after much work I did not figure out how the hints on comments lead to the answer. I tried to use the comparison between spectral radius and operator norm and other elementary facts of spectral theory. More likely I am missing some elementary argument so I decided to ask for some help with the details.