Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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Page 1472
... solution of tσ = λσ square- integrable at a and satisfying the boundary condition B , so is f . Since , by the preceding lemma , only one such solution ( up to a constant multiple ) exists , we must have ƒ = af , where , since | ƒl \ f ...
... solution of tσ = λσ square- integrable at a and satisfying the boundary condition B , so is f . Since , by the preceding lemma , only one such solution ( up to a constant multiple ) exists , we must have ƒ = af , where , since | ƒl \ f ...
Page 1521
... solution of the order of t - 1 - i as t → ∞ and another which behaves like t ' as too . The solution at 2 = 1 — i is exactly similar . Thus , by Theorem XII.4.19 , L1 - 2 has precisely one solution belonging to L2 ( 2 , ∞ ) for each ...
... solution of the order of t - 1 - i as t → ∞ and another which behaves like t ' as too . The solution at 2 = 1 — i is exactly similar . Thus , by Theorem XII.4.19 , L1 - 2 has precisely one solution belonging to L2 ( 2 , ∞ ) for each ...
Page 1529
... solution whose asymp- totic expansion begins with the factor exp ( ¿ 1 ) -1 ) . Thus , a solution ( " small solution " ) with the first kind of asymptotic expansion is uniquely determined by its asymptotic expansion ; while a solution ...
... solution whose asymp- totic expansion begins with the factor exp ( ¿ 1 ) -1 ) . Thus , a solution ( " small solution " ) with the first kind of asymptotic expansion is uniquely determined by its asymptotic expansion ; while a solution ...
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 Haar measure Hence Hilbert space Hilbert-Schmidt operator identity inequality integral interval isometric isomorphism kernel L₁ L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma PROOF prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T transform unique unitary vanishes vector zero