Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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Page 1179
... arbitrary Hilbert space . PROOF . Note that Theorem 20 goes over with trivial modifications of its proof to functions with values in an arbitrary Hilbert space ( or even an arbitrary L - space ) and , in particular , that Lemma 21 ...
... arbitrary Hilbert space . PROOF . Note that Theorem 20 goes over with trivial modifications of its proof to functions with values in an arbitrary Hilbert space ( or even an arbitrary L - space ) and , in particular , that Lemma 21 ...
Page 1337
... arbitrary vector ƒ in L2 ( I ) has an expansion of " Fourier integral " type in terms of eigenfunctions W , ( t , 2 ) of the differential operator v . Unfortunately , the interest of Theorem 1 is more theoretical than practical , since ...
... arbitrary vector ƒ in L2 ( I ) has an expansion of " Fourier integral " type in terms of eigenfunctions W , ( t , 2 ) of the differential operator v . Unfortunately , the interest of Theorem 1 is more theoretical than practical , since ...
Page 1795
... arbitrary even order with variable coefficients . Proc . Nat . Acad . Sci . U.S.A. 38 , 230–235 ( 1952 ) . The Dirichlet and vibration problems for linear elliptic differential equations of arbitrary order . Proc . Nat . Acad . Sci ...
... arbitrary even order with variable coefficients . Proc . Nat . Acad . Sci . U.S.A. 38 , 230–235 ( 1952 ) . The Dirichlet and vibration problems for linear elliptic differential equations of arbitrary order . Proc . Nat . Acad . Sci ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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adjoint extension adjoint operator algebra analytic B-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers continuous function converges Corollary deficiency indices Definition denote dense eigenvalues element equation essential spectrum Exercise exists finite dimensional follows from Lemma follows from Theorem follows immediately formal differential operator formally self adjoint formula Fourier function defined function f Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T transform unique unitary vanishes vector zero