Linear Operators: Spectral theory |
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Page 1310
... solution y ( t , λ ) of ( r − λ ) y = 0 square - integrable at b and satisfying the boundary conditions at b ... solution y of ( 7-2 ) = 0 square - integrable at b and satisfying all boundary conditions at b , and at least one solution ...
... solution y ( t , λ ) of ( r − λ ) y = 0 square - integrable at b and satisfying the boundary conditions at b ... solution y of ( 7-2 ) = 0 square - integrable at b and satisfying all boundary conditions at b , and at least one solution ...
Page 1521
... solution of the order of t - 1 - i as t → ∞ and another which behaves like ti as too . The solution at 1 - i is exactly similar . Thus , by Theorem XII.4.19 , L1 - λ has precisely one solution belonging to L2 ( 2 , ∞ ) for each non ...
... solution of the order of t - 1 - i as t → ∞ and another which behaves like ti as too . The solution at 1 - i is exactly similar . Thus , by Theorem XII.4.19 , L1 - λ has precisely one solution belonging to L2 ( 2 , ∞ ) for each non ...
Page 1553
... solution of the equation ( 2-7 ) 0. Prove : ( a ) either f is square - integrable , or the point 2 belongs to the essential spectrum of 7 ; ( b ) if all solutions of the equation ( 2-7 ) ƒ = 0 are bounded , then belongs to the essential ...
... solution of the equation ( 2-7 ) 0. Prove : ( a ) either f is square - integrable , or the point 2 belongs to the essential spectrum of 7 ; ( b ) if all solutions of the equation ( 2-7 ) ƒ = 0 are bounded , then belongs to the essential ...
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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