Linear Operators: Spectral theory |
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Page 1079
... eigenvalues of 4 ( each eigenvalue being repeated a number of times equal to the dimension of the range of E ( λ ; A ) ) , then the eigenvalues of 4 ( m ) are his hig ... him i1 , 2 , ... , im being an arbitrary sequence of integers ...
... eigenvalues of 4 ( each eigenvalue being repeated a number of times equal to the dimension of the range of E ( λ ; A ) ) , then the eigenvalues of 4 ( m ) are his hig ... him i1 , 2 , ... , im being an arbitrary sequence of integers ...
Page 1383
... eigen- values are consequently to be determined from the equation sin √λ = 0 . Consequently , in Case A , the eigenvalues & are the numbers of the form ( nл ) 2 , n ≥ 1 ; in Case C , the numbers { ( n + 1 ) л } 2 , n ≥ 0. In Case A ...
... eigen- values are consequently to be determined from the equation sin √λ = 0 . Consequently , in Case A , the eigenvalues & are the numbers of the form ( nл ) 2 , n ≥ 1 ; in Case C , the numbers { ( n + 1 ) л } 2 , n ≥ 0. In Case A ...
Page 1497
... eigenvalues which , by Lemma 29 and Corollary 24 , approach plus infinity . By Theorem 64 , we now see that the discrete eigenvalues of those two problems are the only possible end points in the collection of intervals or " bands ...
... eigenvalues which , by Lemma 29 and Corollary 24 , approach plus infinity . By Theorem 64 , we now see that the discrete eigenvalues of those two problems are the only possible end points in the collection of intervals or " bands ...
Contents
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