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For a bounded self-adjoint operator $L$ on a complex Hilbert space, defineThis equals the spectrum of a bounded operator after removing its isolated eigenvalues of finite multiplicity. To see isolation when the kernel is finite and the shifted range is closed, restrict the shifted operator to its kernel complement: it is invertible there by image-kernel orthogonality for an adjoint. A Neumann-series perturbation leaves nearby nonzero shifts invertible on that complement and on the finite kernel. The closed range must concern the shifted operator, not the original one.

For an isomorphism $T$, if $\Vert T^{-1}(S-T)\Vert <1$, thenis invertible by neumann-series perturbation.