Abstract
We consider the operator \(A=S+B,\) where \(S\) is an unbounded normal operator in a separable Hilbert space \(H,\) having a compact inverse one and \(B\) is a linear operator in \(H,\) such that \(BS^{-1} \) is compact. Let \(\{e_k\}_{k=1}^\infty \) be the normalized eigenvectors of \(S\) and \(B\) be represented in \(\{e_k\}_{k=1}^\infty \) by a matrix \((b_{jk})_{j,k=1}^\infty .\) We approximate the eigenvalues of \(A\) by a combination of the eigenvalues of \(S\) and the eigenvalues of the finite matrix \({(b_{jk})}_{j,k=1}^{n}.\) Applications of to differential operators are also discussed.
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