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If I take a infinite-dimensional square matrix, what can I say about its eigenvalue spectrum? Will they have a discrete infinity of eigenvalues or continuous infinity of them?

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marked as duplicate by Chris White, Brandon Enright, John Rennie, Qmechanic Feb 13 at 8:21

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discussed in many other questions, see e.g.: physics.stackexchange.com/q/95193 physics.stackexchange.com/q/68639 (for normalizability) physics.stackexchange.com/q/98462 (for the definition of different bases - defines the infinite dimensional matrix) –  Martin Feb 13 at 7:53

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Infinite matrices, if properly handled, are nothing but linear operators (either bounded or unbounded) on the Hilbert space $\ell^2(\mathbb N)$. So they can have point spectrum, continuous spectrum, residual spectrum just in view of the general theory of operators in general Hilbert spaces.

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