Linear Operators: Spectral theory |
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Page 949
... seen ( Theorem 2 ) that AP is isometric and isomorphic with C ( S ) , where S is a compact Abelian group , and also ( Lemma 3 ) that the con- tinuous characters of S are of the form ei . By Theorem 1.6 , the set of continuous characters ...
... seen ( Theorem 2 ) that AP is isometric and isomorphic with C ( S ) , where S is a compact Abelian group , and also ( Lemma 3 ) that the con- tinuous characters of S are of the form ei . By Theorem 1.6 , the set of continuous characters ...
Page 977
... seen that 2π 1 R G ( u , v ) == ( -iei ) n lim 2π S " [ 2 " f ( r ) rdr ei ( no ' - ( rs sin 0 ' ) ) do ' . R → ∞ Now the Bessel function Jn of order n is defined by the equation hence we have Jn ( z ) = - 2π 1 2πλο ei ( no - z sin 0 ) ...
... seen that 2π 1 R G ( u , v ) == ( -iei ) n lim 2π S " [ 2 " f ( r ) rdr ei ( no ' - ( rs sin 0 ' ) ) do ' . R → ∞ Now the Bessel function Jn of order n is defined by the equation hence we have Jn ( z ) = - 2π 1 2πλο ei ( no - z sin 0 ) ...
Page 1154
... seen from Corollary III.11.6 , is a consequence of the assertion that ( ii ) ; ( 2 ) ( 4xB ) = cλ ( 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 ) ( 4xB ) = cλ ( 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 ...
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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