Linear Operators: Spectral theory |
From inside the book
Results 1-3 of 33
Page 938
... closed in the C ( G ) topology of C * ( G ) . If the function f is in the unit sphere of C ( G ) then —e ( s ) ≤ f ( s ) ≤ e ( s ) and so , for x * in K , −1 ≤ x * f ≤ 1 , which shows that x * ≤ 1. Thus K is a convex subset of the ...
... closed in the C ( G ) topology of C * ( G ) . If the function f is in the unit sphere of C ( G ) then —e ( s ) ≤ f ( s ) ≤ e ( s ) and so , for x * in K , −1 ≤ x * f ≤ 1 , which shows that x * ≤ 1. Thus K is a convex subset of the ...
Page 1000
... closed . The Krein - Smulian theorem shows that it suffices to demonstrate that the intersection of S ( U ) with every positive multiple of the closed unit sphere in L ( -∞ , ∞ ) is L - closed . Since L1 ( -∞ , ∞ ) is separable it ...
... closed . The Krein - Smulian theorem shows that it suffices to demonstrate that the intersection of S ( U ) with every positive multiple of the closed unit sphere in L ( -∞ , ∞ ) is L - closed . Since L1 ( -∞ , ∞ ) is separable it ...
Page 1902
... closed operators , VII.9.4 ( 601 ) Cauchy integral theorem , ( 225 ) Cauchy problem , ( 613–614 ) , ( 639–641 ) ... sphere , II.4.1 ( 70 ) Closed unit sphere , II.3.1 ( 59 ) Closure of a set , criterion to be in , 1.7.2 ( 27 ) definition ...
... closed operators , VII.9.4 ( 601 ) Cauchy integral theorem , ( 225 ) Cauchy problem , ( 613–614 ) , ( 639–641 ) ... sphere , II.4.1 ( 70 ) Closed unit sphere , II.3.1 ( 59 ) Closure of a set , criterion to be in , 1.7.2 ( 27 ) definition ...
Contents
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Copyright | |
59 other sections not shown
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