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 ' m ( 13+ ) 2 ≤ ( ~ n , 2m ) + 1 + 1 ( Zn , Zn - Zm ) + | ≤ | ( Zn , 2m ) ...
... 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 ' m ( 13+ ) 2 ≤ ( ~ n , 2m ) + 1 + 1 ( Zn , Zn - Zm ) + | ≤ | ( Zn , 2m ) ...
Page 1383
... consequently to be determined from the equation sin √λ = 0 . Consequently , in Case A , the eigenvalues λ are the numbers of the form ( nл ) 2 , n ≥ 1 ; in Case C , the numbers { ( n + 1 ) π } 2 , n ≥ 0. In Case A , the ( normalized ) ...
... consequently to be determined from the equation sin √λ = 0 . Consequently , in Case A , the eigenvalues λ are the numbers of the form ( nл ) 2 , n ≥ 1 ; in Case C , the numbers { ( n + 1 ) π } 2 , n ≥ 0. In Case A , the ( normalized ) ...
Page 1387
... Consequently , by Theorem 3.16 , the resolvent R ( λ ; T ) is an integral operator with the kernel sin Vas ( cos Vat + i sin Vat ) √λ s < t , $ 2 > 0 , sin √āt ( cos √ās + i sin √ās ) t < s , $ 2 > 0 , sin Vas ( cos Vit - i sin Vat ) ...
... Consequently , by Theorem 3.16 , the resolvent R ( λ ; T ) is an integral operator with the kernel sin Vas ( cos Vat + i sin Vat ) √λ s < t , $ 2 > 0 , sin √āt ( cos √ās + i sin √ās ) t < s , $ 2 > 0 , sin Vas ( cos Vit - i sin Vat ) ...
Contents
BAlgebras | 859 |
Bounded Normal Operators in Hilbert Space | 887 |
Miscellaneous Applications | 937 |
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adjoint extension adjoint operator algebra analytic B-algebra B*-algebra Borel set boundary conditions boundary values bounded operator C₁ closed closure Co(I coefficients complex numbers converges Corollary deficiency indices Definition denote dense domain eigenfunctions eigenvalues element essential spectrum 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 kernel L₁(R L₂(I L₂(R Lemma Let f linearly independent mapping Math matrix measure neighborhood norm open set operators in Hilbert orthogonal orthonormal partial differential operator Plancherel's theorem positive preceding lemma prove real axis real numbers representation satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose symmetric operator T₁ T₂ theory To(t topology unique unitary vanishes vector zero