Linear Operators: Spectral theory |
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Page 1245
... fact that each complex number a has a unique representation α = α reio , where r≥ 0 , and e1o 1. By analogy with the fact that r = we shall first seek to obtain the self adjoint operator A from the operator T * T . = ( ãα ) 1 , 1 LEMMA ...
... fact that each complex number a has a unique representation α = α reio , where r≥ 0 , and e1o 1. By analogy with the fact that r = we shall first seek to obtain the self adjoint operator A from the operator T * T . = ( ãα ) 1 , 1 LEMMA ...
Page 1348
... fact makes evident what could readily be suspected before : that from the point of view of the negative real axis , a more favorable choice of basis for the solutions of tσ = λσ is T1 τσ ô1 ( t , λ ) = e − t√ ̄Ã ̧ ô2 ( t , λ ) = et ...
... fact makes evident what could readily be suspected before : that from the point of view of the negative real axis , a more favorable choice of basis for the solutions of tσ = λσ is T1 τσ ô1 ( t , λ ) = e − t√ ̄Ã ̧ ô2 ( t , λ ) = et ...
Page 1647
... fact everywhere identical . Thus , by Lemma 3 , a distribution F corresponds to a unique continuous function if it cor- responds to any continuous function at all . The following definition shows the way in which a distribution in I may ...
... fact everywhere identical . Thus , by Lemma 3 , a distribution F corresponds to a unique continuous function if it cor- responds to any continuous function at all . The following definition shows the way in which a distribution in I may ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
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Common terms and phrases
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 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 PROOF prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T unique unitary vanishes vector zero