The spectral theorem says that if $A$ is a self-adjoint operator on a Hilbert space $H$ with compact inverse, then the eigenvectors of $A$ form a complete orthonormal basis of $H$. Furthermore each eigenvalue is real and $|\lambda_n|\to\infty$ as $n\to\infty$.
Now let us slightly change the conditions of the theorem. Let $A$ be a self-adjoint operator with increasing sequence of eigenvalues $0<\lambda_1<\lambda_2<...$ Is it true that the eigenvectors of $A$ form a complete orthonormal basis of $H$?