Linear Operators, Part 2Interscience Publishers, 1963 - Algebra, Universal |
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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 ---- ( | 3n | + ) 2 ≤ | ( Zn , ≈m ) + | + | ( ≈n , Zn − Zm ) + ...
... 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 ---- ( | 3n | + ) 2 ≤ | ( Zn , ≈m ) + | + | ( ≈n , Zn − Zm ) + ...
Page 1383
... consequently to be determined from the equation sin√ = 0 . Consequently , in Case A , the eigenvalues λ are the numbers of the form ( në ) 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√ = 0 . Consequently , in Case A , the eigenvalues λ are the numbers of the form ( në ) 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 Vās ( cos Vāt + i sin √āt ) s < t , $ 2 > 0 , sin Vat ( cos Vis + i sin Vās ) t < s , $ 20 , sin √ās ( cos √āt — i sin √āt ) s ...
... Consequently , by Theorem 3.16 , the resolvent R ( 2 ; T ) is an integral operator with the kernel sin Vās ( cos Vāt + i sin √āt ) s < t , $ 2 > 0 , sin Vat ( cos Vis + i sin Vās ) t < s , $ 20 , sin √ās ( cos √āt — i sin √āt ) s ...
Contents
IX | 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 compact subset complex numbers continuous function converges Corollary deficiency indices Definition denote dense domain 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 isometric isomorphism kernel L₁(R L₂(I L₂(R Lemma Let f linear linearly independent mapping matrix measure neighborhood norm open set open subset orthonormal partial differential operator Plancherel's theorem positive PROOF prove real axis real numbers satisfies Section sequence solution spectral spectral theory square-integrable subspace Suppose T₁ T₁(t theory To(t topology unique unitary vanishes vector zero