Linear Operators: Spectral operators |
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Page 1378
... Theorem 13. In particular , i is unique . Thus , all that remains for us to prove is that if p ;; ( e ) = 0 for j > k , then σ1 , . . . , σ is a determining set for T. Suppose that i 1 , . . . , m , be as in Theorem 01 , this is not the ...
... Theorem 13. In particular , i is unique . Thus , all that remains for us to prove is that if p ;; ( e ) = 0 for j > k , then σ1 , . . . , σ is a determining set for T. Suppose that i 1 , . . . , m , be as in Theorem 01 , this is not the ...
Page 1379
... THEOREM . Let v , T , A , 01 , ... , σ , etc. , be as in Theorem 18 . Then if , for j > k , the functions 0 ;; ( λ ) of Theorem 18 ( or , the functions 05 ( 2 ) of Theorem 18 ) may be extended to analytic functions defined on the whole ...
... THEOREM . Let v , T , A , 01 , ... , σ , etc. , be as in Theorem 18 . Then if , for j > k , the functions 0 ;; ( λ ) of Theorem 18 ( or , the functions 05 ( 2 ) of Theorem 18 ) may be extended to analytic functions defined on the whole ...
Page 1904
... theorems , .IV.15 Alexandroff theorem on conver- IV.9.15 gence of measures , ( 316 ) Arzelą theorem on continuous limits , IV.6.11 ( 268 ) Banach theorem for operators into space of measurable functions , IV.11.2-8 ( 332–333 ) Egoroff ...
... theorems , .IV.15 Alexandroff theorem on conver- IV.9.15 gence of measures , ( 316 ) Arzelą theorem on continuous limits , IV.6.11 ( 268 ) Banach theorem for operators into space of measurable functions , IV.11.2-8 ( 332–333 ) Egoroff ...
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