Linear Operators: Spectral theory |
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Page 1447
... real . Hence , by Theorem 5 , σ , ( 7 ) is contained in the real axis . By the previous theorem the polynomial P ( t ) = Σ - 09 ; ( it ) 3 is real . If n is odd , this polynomial is of odd order . Hence , it converges to + ∞ as t ...
... real . Hence , by Theorem 5 , σ , ( 7 ) is contained in the real axis . By the previous theorem the polynomial P ( t ) = Σ - 09 ; ( it ) 3 is real . If n is odd , this polynomial is of odd order . Hence , it converges to + ∞ as t ...
Page 1493
... real axis . Hence we must have m = 1 , so that 1 is one - to - one in the neighborhood of λ = 2 , and the inverse image under 9 , of a small arc of the unit circle containing 1 ( 2 ) is a small arc of the real axis containing 2 . Since ...
... real axis . Hence we must have m = 1 , so that 1 is one - to - one in the neighborhood of λ = 2 , and the inverse image under 9 , of a small arc of the unit circle containing 1 ( 2 ) is a small arc of the real axis containing 2 . Since ...
Page 1610
... real axis ( Naimark [ 5 ] ) . Other conditions which allow approximate determination of the essential spectrum are the following : ( 14 ) Suppose that has the form given in ( 4 ) and that all coefficients are real and eventually non ...
... real axis ( Naimark [ 5 ] ) . Other conditions which allow approximate determination of the essential spectrum are the following : ( 14 ) Suppose that has the form given in ( 4 ) and that all coefficients are real and eventually non ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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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 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 prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology transform unique unitary vanishes vector zero