On the removal of finite discrete spectrum by coefficient stripping (Q1933406)

The main goal of this paper is to prove the following result. Let \(J\) be a bounded self-adjoint operator on a Hilbert space \(\mathcal{H}\), and let \((P_{n}), (Q_{n}), (R_{n})\) be sequences of orthogonal projections in \(\mathcal{H}\) satisfying \[ P_{n} Q_{n} = Q_{n} P_{n} = P_{n} R_{n} = R_{n} P_{n} = Q_{n} R_{n} = R_{n} Q_{n} = 0, \] \[ P_{n} + Q_{n} + R_{n} = I, \] \[ s - \lim_{n \to \infty} Q_{n} = s - \lim_{n \to \infty} P_{n} = 0, \] \[ P_{n} J Q_{n} = Q_{n} J P_{n} = 0. \] It is stated that, if the spectrum of \(J\) on the interval \((- \infty, 0)\) consists only of finitely many eigenvalues of finite multiplicity, then, for sufficiently large \(n\), \(Q _{n} J Q_{n}\) is a non-negative operator. The result is applied to block Jacobi matrices and also to Sturm-Liouville operators with operator-valued coefficients.