Linear Operators, Part 2 |
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Page 1036
... converges absolutely provided that λ for any k . In view of the fact that → 0 it follows from the estimate in ( * ) that the series . ∞ Σ 2 log ( ex ( 1-4 ) ) k = 1 λ converges uniformly and absolutely for each compact set of numbers ...
... converges absolutely provided that λ for any k . In view of the fact that → 0 it follows from the estimate in ( * ) that the series . ∞ Σ 2 log ( ex ( 1-4 ) ) k = 1 λ converges uniformly and absolutely for each compact set of numbers ...
Page 1436
... converges , then { f } converges . Let { g } be a bounded sequence of elements of D ( T ) such that { Tgn } converges . Find a subsequence { gn , } = { h } such that x * ( h , ) converges for each j , 1 ≤i≤k . Then h1 = h ̧ - Σ_x ( h ...
... converges , then { f } converges . Let { g } be a bounded sequence of elements of D ( T ) such that { Tgn } converges . Find a subsequence { gn , } = { h } such that x * ( h , ) converges for each j , 1 ≤i≤k . Then h1 = h ̧ - Σ_x ( h ...
Page 1664
... converges unconditionally to F. PROOF . It follows from the Definition 37 of the topology in D ( C ) that it suffices to show that ( 2π ) " ΣΕ L eiLq ( x ) dx L converges unconditionally to F ( p ) for each q in Ca ( C ) . For any set A ...
... converges unconditionally to F. PROOF . It follows from the Definition 37 of the topology in D ( C ) that it suffices to show that ( 2π ) " ΣΕ L eiLq ( x ) dx L converges unconditionally to F ( p ) for each q in Ca ( C ) . For any set A ...
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 domain 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 linearly independent mapping matrix measure neighborhood non-zero norm open set operators in Hilbert orthogonal orthonormal basis Plancherel's theorem positive preceding lemma prove real axis real numbers satisfies sequence solution spectral spectral theorem square-integrable subspace Suppose T₁ T₂ theory To(t topology tr(T transform unique unitary vanishes vector zero