Linear Operators: Spectral theory |
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Page 1215
... preceding proof we let F that , by Lemma 9 , n [ TM , ( Uƒ ) a ( 2 ) Wa ( ' , 2 ) μa ( d2 ) → F ( T ) a = = E ( -n , n ] ) F ( T ) a UZ1F = fa - ( Uf ) a so Thus the integral ƒ ‰ ( Uƒ ) , ( 2 ) W。( s , λ ) μ 。( d2 ) exists in the ...
... preceding proof we let F that , by Lemma 9 , n [ TM , ( Uƒ ) a ( 2 ) Wa ( ' , 2 ) μa ( d2 ) → F ( T ) a = = E ( -n , n ] ) F ( T ) a UZ1F = fa - ( Uf ) a so Thus the integral ƒ ‰ ( Uƒ ) , ( 2 ) W。( s , λ ) μ 。( d2 ) exists in the ...
Page 1378
... preceding theorem . Moreover , in the course of the proof preceding the statement of Theorem 23 , it was shown that if 2 is any point in △ , there exists a small open subinterval N of △ , containing 2 , such that the set of ...
... preceding theorem . Moreover , in the course of the proof preceding the statement of Theorem 23 , it was shown that if 2 is any point in △ , there exists a small open subinterval N of △ , containing 2 , such that the set of ...
Page 1419
... preceding lemma , f ( t ) ≤ fi ( t ) in [ 8 ; +1 , mi + 1 ] . In particular —f ( m ; +1 ) = | f ( m ; +1 ) ¦ ≤ f1 ... preceding corollary . Q.E.D. PROOF OF THEOREM 24. If the function q of Theorem 24 is not negative for t sufficiently ...
... preceding lemma , f ( t ) ≤ fi ( t ) in [ 8 ; +1 , mi + 1 ] . In particular —f ( m ; +1 ) = | f ( m ; +1 ) ¦ ≤ f1 ... preceding corollary . Q.E.D. PROOF OF THEOREM 24. If the function q of Theorem 24 is not negative for t sufficiently ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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A₁ 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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero