Linear Operators in Hilbert Spaces

adjoint analytic vectors assertion follows Assume Auxiliary theorem B(H₁ bijective Carleman operator Cauchy sequence closable complex Hilbert space Consequently continuous convergent countable defect indices denote densely defined differential ED(T eigenvalue Ep(T equality essentially self-adjoint everywhere EXAMPLE exists a sequence f₁ finite formula function f ƒ ² ƒ E D(T ƒ E H ƒ EH H₁ H₁ and H₂ H₁ into H₂ H₂ Hence Hilbert-Schmidt operator Hint implies integral K-real L₂(a L₂(M L₂(R Let ƒ Let H Levi's theorem M₁ M₂ measurable function non-negative norm numbers obviously operator from H operator on H orthogonal projection orthonormal orthonormal basis p-measurable P₁ P₂ pre-Hilbert space PROOF prove scalar product self-adjoint extensions self-adjoint operator sequence f sesquilinear form space H spectral family subset symmetric operator T-bounded T₁ T₂ unitary W₂