Linear Operators: Spectral theory |
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Page 1036
... converges absolutely provided that λ λx for any k . In view of the fact that ( * ) that the series ∞ k - 1 → 0 it follows from the estimate in log ( et ( 1-4 ) ) converges uniformly and absolutely for each compact set of numbers λ ...
... converges absolutely provided that λ λx for any k . In view of the fact that ( * ) that the series ∞ k - 1 → 0 it follows from the estimate in log ( et ( 1-4 ) ) converges uniformly and absolutely for each compact set of numbers λ ...
Page 1333
... converge to zero in L2 ( I ) . Thus by Lemma 2.16 the series ( fi ) i converges to ƒ in the topology of C - 1 ( J ) for each compact interval J of I. Since the series converges unconditionally in L2 ( I ) , it follows that it converges ...
... converge to zero in L2 ( I ) . Thus by Lemma 2.16 the series ( fi ) i converges to ƒ in the topology of C - 1 ( J ) for each compact interval J of I. Since the series converges unconditionally in L2 ( I ) , it follows that it converges ...
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 * ( h1 ) converges for each j , 1 ≤i≤k . Then h ; = h1 - Σ_x * ( h1 ) ...
... 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 * ( h1 ) converges for each j , 1 ≤i≤k . Then h ; = h1 - Σ_x * ( h1 ) ...
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