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We improve known perturbation results for self-adjoint operators in Hilbertspaces and prove spectral enclosures for diagonally dominant $J$-self-adjointoperator matrices. These are used in the proof of the central result, aperturbation theorem for $J$-non-negative operators. The results are applied tosingular indefinite Sturm-Liouville operators with $L^p$-potentials. Knownbounds on the non-real eigenvalues of such operators are improved.

Relatively bounded perturbations of J-non-negative operators