## Linear Operators in Hilbert Spaces |

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adjoint arbitrary assertion follows Assume assumption Banach space belongs bounded called Carleman operator Cauchy sequence choose closable closed compact complex Hilbert space Consequently consider contains continuous convergent corresponding countable defined definition denote dense densely defined differential ED(T eigenvalue elements equality equation equivalent essentially self-adjoint everywhere EXAMPLE Exercise exists exists a sequence fED(T finite follows formula function given Hence holds implies induced inequality integral interval Let H limit linear mapping maximal measurable multiplication non-negative norm obtain obviously operator from H orthogonal projection orthonormal p-measurable particular positive pre-Hilbert space PROOF properties Proposition prove respectively satisfied scalar product self-adjoint extensions self-adjoint operator separable space H spectral family spectrum step subset subspace symmetric operator T-bounded Theorem uniquely unitary write zero