Linear Operators, Part 2 |
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Page 977
... seen that 1 R 2π G ( u , v ) == 2 - ( - ie ) " lim [ " f ( r ) rdr [ * " R - ∞ 2π ei ( no ' - ( rs sin 0 ' ) ) do ' . Now the Bessel function Jn of order n is defined by the equation hence we have 1 2π Jn ( z ) = ei ( no - z sin 0 ) do ...
... seen that 1 R 2π G ( u , v ) == 2 - ( - ie ) " lim [ " f ( r ) rdr [ * " R - ∞ 2π ei ( no ' - ( rs sin 0 ' ) ) do ' . Now the Bessel function Jn of order n is defined by the equation hence we have 1 2π Jn ( z ) = ei ( no - z sin 0 ) do ...
Page 1037
... seen that the function ( T ) is analytic for λ 0 and vanishes only for 2 in σ ( T ) . It remains to show that if 2 0 , then ( T ) is continuous in T relative to the Hilbert - Schmidt norm in HS . To do this let { T } be a sequence in HS ...
... seen that the function ( T ) is analytic for λ 0 and vanishes only for 2 in σ ( T ) . It remains to show that if 2 0 , then ( T ) is continuous in T relative to the Hilbert - Schmidt norm in HS . To do this let { T } be a sequence in HS ...
Page 1154
... seen from Corollary III.11.6 , is a consequence of the assertion that 2 ( 2 ) ( A × B ) = cλ ( A ) 2 ( B ) , ( ii ) Α , Β Ε Σ . Thus we shall endeavor to establish ( ii ) . For every E in ( 2 ) let μ ( E ) = 2 ( 2 ) ( hE ) where h is ...
... seen from Corollary III.11.6 , is a consequence of the assertion that 2 ( 2 ) ( A × B ) = cλ ( A ) 2 ( B ) , ( ii ) Α , Β Ε Σ . Thus we shall endeavor to establish ( ii ) . For every E in ( 2 ) let μ ( E ) = 2 ( 2 ) ( hE ) where h is ...
Contents
BAlgebras | 861 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure coefficients compact operator complex numbers constant 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 prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology unique unitary vanishes vector zero