Let $T$ and $S$ be two symmetric, compact linear operators on a (separable) Hilbert space $H$ that commute. Why is there at least 1 common eigenvector of $T$ and $S$?
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Two commuting compact operators are simultaneously diagonalizable. This means that there exists an orthonormal basis such that both $S$ and $T$ are diagonal with respect to that basis, i.e. every vector in such a basis is a common eigenvector.