Linear Operators: Spectral theory |
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Page 1383
... eigenfunctions being complete . With boundary condi- tions A and C , the unique solution of 73σ = λo satisfying the boundary condition 73σ = 2o is sin √t . With boundary conditions A , the eigen- values are consequently to be ...
... eigenfunctions being complete . With boundary condi- tions A and C , the unique solution of 73σ = λo satisfying the boundary condition 73σ = 2o is sin √t . With boundary conditions A , the eigen- values are consequently to be ...
Page 1386
... eigenfunctions of a differential operator can be obtained directly from the Titchmarsh - Kodaira theorem . In the present case , e - k * is easy enough to normalize directly , but in those cases to be studied below , in which the ...
... eigenfunctions of a differential operator can be obtained directly from the Titchmarsh - Kodaira theorem . In the present case , e - k * is easy enough to normalize directly , but in those cases to be studied below , in which the ...
Page 1582
... eigenfunctions of a differential operator on a closed interval , such as the existence of an infinite sequence of eigenvalues with no finite cluster point , the orthogonality property of the eigen- functions and the Parseval equality ...
... eigenfunctions of a differential operator on a closed interval , such as the existence of an infinite sequence of eigenvalues with no finite cluster point , the orthogonality property of the eigen- functions and the Parseval equality ...
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