Linear Operators: Spectral theory |
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Page 890
... formula ( vi ) , but in this situation it is necessary to define the integral appearing in ( vi ) and to define the algebra of scalar functions f to which the formula may be applied . One class of scalar functions f , other than ...
... formula ( vi ) , but in this situation it is necessary to define the integral appearing in ( vi ) and to define the algebra of scalar functions f to which the formula may be applied . One class of scalar functions f , other than ...
Page 1112
... formula and Cramer's formula for matrix inverses , we have d dz d d ΣΥ det ( A + zB ) \ 2 = 0 = Σ Σbi Yu i = 1 j = 1 = det ( 4 ) tr ( 4 - 1B ) , where Yij denotes the cofactor of the element a ,, of the matrix A. Substituting A = I + T ...
... formula and Cramer's formula for matrix inverses , we have d dz d d ΣΥ det ( A + zB ) \ 2 = 0 = Σ Σbi Yu i = 1 j = 1 = det ( 4 ) tr ( 4 - 1B ) , where Yij denotes the cofactor of the element a ,, of the matrix A. Substituting A = I + T ...
Page 1363
... formula E ( ( 1 , 2 ) ) = lim lim + 0-3 0-8 1 Σπί 21 + 8 [ R ( 1 - iɛ ; T ) -R ( λ + iɛ ; T ) ] fdλ . The problem we face is that of passing from this latter formula in- volving the resolvent to a formula involving the individual terms ...
... formula E ( ( 1 , 2 ) ) = lim lim + 0-3 0-8 1 Σπί 21 + 8 [ R ( 1 - iɛ ; T ) -R ( λ + iɛ ; T ) ] fdλ . The problem we face is that of passing from this latter formula in- volving the resolvent to a formula involving the individual terms ...
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