Linear Operators: Spectral theory |
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Page 1245
... fact that each complex number & has a unique representation α = re1o , where r≥ 0 , and | ei | = 1. By analogy with the fact that r = ( a ) we shall first seek to obtain the self adjoint operator A from the operator TT . α 1 LEMMA ...
... fact that each complex number & has a unique representation α = re1o , where r≥ 0 , and | ei | = 1. By analogy with the fact that r = ( a ) we shall first seek to obtain the self adjoint operator A from the operator TT . α 1 LEMMA ...
Page 1348
... fact entire in 2. In the range suitable basis for the solutions of to λ < 0 , 2 is imaginary , and an analytic expression like cost is hard to work with because of the apparent ambiguity in the definition of 2. An analysis of the ...
... fact entire in 2. In the range suitable basis for the solutions of to λ < 0 , 2 is imaginary , and an analytic expression like cost is hard to work with because of the apparent ambiguity in the definition of 2. An analysis of the ...
Page 1386
... fact has already been noted ; but the present derivation is of particular interest since it serves to emphasize the fact that the normalization factors for the orthonormal eigenfunctions of a differential operator can be obtained ...
... fact has already been noted ; but the present derivation is of particular interest since it serves to emphasize the fact that the normalization factors for the orthonormal eigenfunctions of a differential operator can be obtained ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 PROOF prove real axis satisfies sequence solution spectral spectral theorem square-integrable subset subspace Suppose T₁ T₁(t T₂ theory To(t topology tr(T unique unitary vanishes vector zero