Linear Operators, Part 2 |
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Page 919
... equivalent operators have identical properties in H. 1 12 THEOREM . Two bounded normal operators in a separable Hilbert space Hare unitarily equivalent if and only if the corresponding ordered representations of relative to the ...
... equivalent operators have identical properties in H. 1 12 THEOREM . Two bounded normal operators in a separable Hilbert space Hare unitarily equivalent if and only if the corresponding ordered representations of relative to the ...
Page 920
... equivalent . Let E and È be the resolutions of the identity for T and Ĩ respectively . From Corollary 2.7 it is seen ... equivalent . To prove the converse it is assumed that U and Ũ are equivalent . By Lemma 11 there is an isometry Vn ...
... equivalent . Let E and È be the resolutions of the identity for T and Ĩ respectively . From Corollary 2.7 it is seen ... equivalent . To prove the converse it is assumed that U and Ũ are equivalent . By Lemma 11 there is an isometry Vn ...
Page 1217
... equivalent to U. More- over two self adjoint operators in H are unitarily equivalent if and only if the corresponding ordered representations of § relative to the operators are equivalent . PROOF . Let E , E1 be the resolutions of the ...
... equivalent to U. More- over two self adjoint operators in H are unitarily equivalent if and only if the corresponding ordered representations of § relative to the operators are equivalent . PROOF . Let E , E1 be the resolutions of the ...
Contents
BAlgebras | 861 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers constant 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 Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linear linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology unique unitary vanishes vector zero