Linear Operators: Spectral theory |
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Page 977
... seen that 1 R 2π G ( u , v ) - ( -iei ) n lim f ( r ) rdr e1 ( no ' — ( rs sin 0 ′ ) ) do ' . 2π ∞4 - y 0 Now the Bessel function Jn of order n is defined by the equation Jn ( z ) - 1 2π . 2π ei ( no - z sin 0 ) do ; hence we have R G ...
... seen that 1 R 2π G ( u , v ) - ( -iei ) n lim f ( r ) rdr e1 ( no ' — ( rs sin 0 ′ ) ) do ' . 2π ∞4 - y 0 Now the Bessel function Jn of order n is defined by the equation Jn ( z ) - 1 2π . 2π ei ( no - z sin 0 ) do ; hence we have R G ...
Page 1154
... seen from Corollary III.11.6 , is a consequence of the assertion that ( ii ) 2 ( 2 ) ( A × B ) = c2 ( A ) 2 ( B ) , Α , Β Ε Σ . Thus we shall endeavor to establish ( ii ) . For every E in Σ ( 2 ) let μ ( E ) 2 ( 2 ) ( hE ) where h is ...
... seen from Corollary III.11.6 , is a consequence of the assertion that ( ii ) 2 ( 2 ) ( A × B ) = c2 ( A ) 2 ( B ) , Α , Β Ε Σ . Thus we shall endeavor to establish ( ii ) . For every E in Σ ( 2 ) let μ ( E ) 2 ( 2 ) ( hE ) where h is ...
Page 1324
... seen ( cf. Theorem 10 ) that ta ; 0 , i - - 1 , ... , n . Thus choosing a basis { § } , i ******* 1 , ... , n , for the solutions of to = 0 , and defining the matrix { T } by the equations τσ n ai Σ Γυξη , i = 1 , j • " n , j = 1 the ...
... seen ( cf. Theorem 10 ) that ta ; 0 , i - - 1 , ... , n . Thus choosing a basis { § } , i ******* 1 , ... , n , for the solutions of to = 0 , and defining the matrix { T } by the equations τσ n ai Σ Γυξη , i = 1 , j • " n , j = 1 the ...
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 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 open set 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 tr(T unique unitary vanishes vector zero