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analytic vectors arbitrary assertion follows Assume Auxiliary theorem Banach space bijective bounded interval Carleman operator Cauchy sequence closable closed subspace compact operator complex Hilbert space Consequently convergent countable defect indices denote densely defined eigenelement eigenvalue elements equality equivalent essentially self-adjoint everywhere Example Exercise exists a sequence finite dimensional follows from Theorem formula G D(T Hence Hermitian Hilbert-Schmidt operator Hint holds implies inequality integral isomorphism Let H Levi's theorem linear operator mapping measurable function non-decreasing non-negative normal operator normed space numbers obviously operator from H operator of multiplication operator on H orthogonal projection orthonormal basis p-measurable pre-Hilbert space Proposition respectively S(Rm scalar product Section self-adjoint extensions self-adjoint operator sesquilinear form space H spectral family spectrum step functions subset subspace of H symmetric operator T-bounded vector space zero