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Commutative B Algebras
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator closed closure coefficients compact subset complex numbers constant continuous function converges Corollary countably deficiency indices Definition denote dense domain eigenfunctions eigenvalues element essential spectrum exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function g Haar measure Hence Hermitian Hilbert space Hilbert-Schmidt operator hypothesis identity inequality infinity integral interval inverse kernel Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm one-to-one open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation restriction satisfies Section sequence set of boundary shows singular solution spectral spectral theorem square-integrable subspace Suppose symmetric operator theory topology transform uniformly unique unitary vanishes vector zero