Spectra of finite and infinite Jordan blocks

The eigenvalues of Jordan blocks are very sensitive to perturbations. This is known for a long time, but why the eigenvalues of a single Jordan block converge to the spectrum of the shift operator when the dimension runs to infinity, is unknown. In this thesis we show why Jordan blocks are so sensitive to perturbations, what has been studied about them in the literature and what the location of the eigenvalues is after perturbation. We also study the shift operator, calculate its spectrum and show that this spectrum is not sensitive to perturbations. Important to note is that the shift operator can be seen as a single Jordan block on an infinite space. We did not find a definite answer to the relation between the two, but by studying the pseudospectra of both the matrix and the operator we give some clues on why the spectrum of both structures are related.