Linear Operators: Spectral theory |
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Page 1174
... f ( s ) ds = [ h ( s ) \ f ( s ) | \ ds ; hence ( 42 ) follows from the similar well - known equation for scalar- valued functions . This concludes the proof of Lemma 21 , and with it the proof of Theorem 20. Q.E.D. 22 COROLLARY . Let ...
... f ( s ) ds = [ h ( s ) \ f ( s ) | \ ds ; hence ( 42 ) follows from the similar well - known equation for scalar- valued functions . This concludes the proof of Lemma 21 , and with it the proof of Theorem 20. Q.E.D. 22 COROLLARY . Let ...
Page 1649
... let F be a distribution in I. Then the distribution Fin I defined by the equation F ( q ) = F ( 9 ) , q Є Co ° ( I ) , is called the complex conjugate of F. 8 LEMMA . Let I and F be as in the preceding definition , and let T be a formal ...
... let F be a distribution in I. Then the distribution Fin I defined by the equation F ( q ) = F ( 9 ) , q Є Co ° ( I ) , is called the complex conjugate of F. 8 LEMMA . Let I and F be as in the preceding definition , and let T be a formal ...
Page 1696
... F to a distribution in D1 ( D ) such that the carrier of G is C , and a sequence of elements Îm in Co ( D ) such that m → G as m → ∞o . Let y be in Co ° ( I ) and have a neighborhood of C. Then my is in G as moo . If m - y ( x ) 1 ...
... F to a distribution in D1 ( D ) such that the carrier of G is C , and a sequence of elements Îm in Co ( D ) such that m → G as m → ∞o . Let y be in Co ° ( I ) and have a neighborhood of C. Then my is in G as moo . If m - y ( x ) 1 ...
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