Linear Operators: Spectral theory |
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Page 1051
... evident from Definition 1. Statements ( ii ) and ( iii ) are evident consequences of Definition 1 and of the formulae √gn P ( x ) dx = √gn P ( Ux ) dx , Jgn P ( xx ) dx = lal Sgn P ( x ) dx , En which are valid for every Lebesgue ...
... evident from Definition 1. Statements ( ii ) and ( iii ) are evident consequences of Definition 1 and of the formulae √gn P ( x ) dx = √gn P ( Ux ) dx , Jgn P ( xx ) dx = lal Sgn P ( x ) dx , En which are valid for every Lebesgue ...
Page 1347
... evident advantage that it makes o , ( , λ ) an entire function of the complex variable 2 ; but it has drawbacks which , though less evident , are nevertheless decisive . Suppose , for example , that we study the self adjoint operator T ...
... evident advantage that it makes o , ( , λ ) an entire function of the complex variable 2 ; but it has drawbacks which , though less evident , are nevertheless decisive . Suppose , for example , that we study the self adjoint operator T ...
Page 1695
... evident that the series ∞ y ( x ) = ΣΥ ; ( 2 ) j = 1 converges to a function y in C ( I ) which is everywhere positive . Thus , if we put n , ( x ) y ( x ) -1y , ( x ) , we have 7 , in Co ( I ) , and = ∞ Ση ; ( 2 ) = 1 . j = 1 Since ...
... evident that the series ∞ y ( x ) = ΣΥ ; ( 2 ) j = 1 converges to a function y in C ( I ) which is everywhere positive . Thus , if we put n , ( x ) y ( x ) -1y , ( x ) , we have 7 , in Co ( I ) , and = ∞ Ση ; ( 2 ) = 1 . j = 1 Since ...
Contents
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Commutative BAlgebras | 874 |
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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 countably deficiency indices Definition denote dense eigenfunctions eigenvalues element equation essential spectrum Exercise exists f₁ finite dimensional follows from Lemma follows from Theorem formal differential operator formally self adjoint formula Fourier function f Haar measure Hence Hilbert space Hilbert-Schmidt operator homomorphism identity inequality infinity integral interval kernel L₁ L₁(R L₂ L₂(I L₂(R Lemma Let f linearly independent mapping matrix measure neighborhood non-zero norm operators in Hilbert orthogonal Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence shows solution spectral set spectral theorem square-integrable subset subspace Suppose T₁ T₂ theory To(t topology tr(T transform uniformly unique unitary vanishes vector zero