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A_lB allowable coefficient analytic arbitrarily small assume assumptions bounded on L2 bounded operator Chapter closure compact operator constant continuously embedded D(An defined denote dense determined differential operators domain e p(A,B eigenfunction eigenvectors equivalent finite sums follows formal operator Fredholm operators H0 to H hence Hilbert space Hn_m implies inner product integer invertible operator last section Lemma 16 Lemma 9 Let A,B linear combination linearly independent matrix multiple non-zero eigenvalues norm nullspace one-to-one operator on L2 order at least pair A,B perturbation perturbation theory pole of RA problem is regular projection associated proof R(A_IB R(EA R(EN R(Eoo R(Pz range satisfies the hypotheses scalar self—adjoint seminorm separated boundary conditions sequence set of boundary simple pole spectral and complete spectral operator subspace super-regular suppose term of order theory topology uniformly bounded vector zeros of M(u