Eigenvalues of Infinite Dimensional Matrix

Question

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?

Answer 1

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.