Linear Operators: Spectral theory |
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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 are unitarily equivalent if and only if the corresponding ordered representations of H relative to the ...
... equivalent operators have identical properties in H. 1 12 THEOREM . Two bounded normal operators in a separable Hilbert space are unitarily equivalent if and only if the corresponding ordered representations of H 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 . n n To prove the converse it is assumed that U and Ũ are equivalent . By Lemma 11 there is an isometry ...
... equivalent . Let E and È be the resolutions of the identity for T and Ĩ respectively . From Corollary 2.7 it is seen ... equivalent . n n To prove the converse it is assumed that U and Ũ are equivalent . By Lemma 11 there is an isometry ...
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 | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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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 Co(I coefficients complex numbers converges Corollary 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 f Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero