Linear Operators: Spectral operators |
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Page 912
... separable then the measure space ( S , E , μ ) in Corollary 4 may be taken to be finite . = PROOF . If is separable then there are a countable number of mutually orthogonal admissible spaces , n = 1 , 2 , . . . , which span H. It will ...
... separable then the measure space ( S , E , μ ) in Corollary 4 may be taken to be finite . = PROOF . If is separable then there are a countable number of mutually orthogonal admissible spaces , n = 1 , 2 , . . . , which span H. It will ...
Page 928
... separable Hilbert space , and let T be a bounded operator which commutes with every operator which commutes with A. Then there exists a bounded measurable function f such that T = f ( A ) . This theorem was stated explicitly by F. Riesz ...
... separable Hilbert space , and let T be a bounded operator which commutes with every operator which commutes with A. Then there exists a bounded measurable function f such that T = f ( A ) . This theorem was stated explicitly by F. Riesz ...
Page 1918
... Separability and metrizability , V.5.1-2 ( 426 ) Separable linear manifolds , II.1.5 ( 50 ) . ( See also Separable sets ) in C , IV.13.16 ( 340 ) in Lp , III.8.5 ( 168 ) , III.9.6 ( 169 ) Separable sets , 1.6.11 ( 21 ) . ( See also ...
... Separability and metrizability , V.5.1-2 ( 426 ) Separable linear manifolds , II.1.5 ( 50 ) . ( See also Separable sets ) in C , IV.13.16 ( 340 ) in Lp , III.8.5 ( 168 ) , III.9.6 ( 169 ) Separable sets , 1.6.11 ( 21 ) . ( See also ...
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 linear 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 satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology unique unitary vanishes vector zero