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Under what circumstances is a (linear) operator [tex] \mathcal{H} \to \mathcal{H}[/tex] between a Hilbert space and itself diagonalizable? Under what circumstances does (number of distinct eigenvalues = dimension of H), i.e., there exists a basis of eigenvectors with distinct eigenvalues? Although I am interested in the answer to these questions from a mathematical point of view, I am thinking about them in the context of QM. If it makes the answer simpler, you can assume the operator is Hermitian.