Linear Operators: Spectral theory |
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Page 1241
... Consequently there is a number M such that M , m = 1 , 2 , . . . . Moreover , given ɛ > 0 there is an integer N such that if m , n > N , then m ― n < ɛ . Thus n → ∞ n ( | ≈n | + ) 2 ≤ | ( Zn , Zm ) + | + | ( Zn , Zn − ≈m ) + ...
... Consequently there is a number M such that M , m = 1 , 2 , . . . . Moreover , given ɛ > 0 there is an integer N such that if m , n > N , then m ― n < ɛ . Thus n → ∞ n ( | ≈n | + ) 2 ≤ | ( Zn , Zm ) + | + | ( Zn , Zn − ≈m ) + ...
Page 1383
... 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 , the ( normalized ) ...
... 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 , the ( normalized ) ...
Page 1387
... Consequently , by Theorem 3.16 , the resolvent R ( 2 ; T ) is an integral operator with the kernel sin √ās ( cos √āt + i sin √āt ) s < t , $ 2 > 0 , sin √āt ( cos √ās + i sin √ās ) √λ t < s , Iλ > 0 , sin √ās ( cos Vīt — i sin ...
... Consequently , by Theorem 3.16 , the resolvent R ( 2 ; T ) is an integral operator with the kernel sin √ās ( cos √āt + i sin √āt ) s < t , $ 2 > 0 , sin √āt ( cos √ās + i sin √ās ) √λ t < s , Iλ > 0 , sin √ās ( cos Vīt — i sin ...
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