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 ∞ log ( eti ( 1-34 ) ) k = 1 λ converges uniformly and absolutely for each compact set of numbers λ ...
... 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 ∞ log ( eti ( 1-34 ) ) k = 1 λ converges uniformly and absolutely for each compact set of numbers λ ...
Page 1420
... converges to zero in the topology of D ( T1 ( T + T ' ) ) . Conversely , let { f } converge to zero in the topology ... converges to zero in and is bounded in D ( T1 ( t ) ) . By hypothesis ( b ) it follows that T1 ( t ' ) hn , converges ...
... converges to zero in the topology of D ( T1 ( T + T ' ) ) . Conversely , let { f } converge to zero in the topology ... converges to zero in and is bounded in D ( T1 ( t ) ) . By hypothesis ( b ) it follows that T1 ( t ' ) hn , converges ...
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 ( 27 ) " ΣΕ L ei L · x q ( x ) dx C converges unconditionally to F ( q ) for each q in Ca ( C ) . For any ...
... converges unconditionally to F. PROOF . It follows from the Definition 37 of the topology in D ( C ) that it suffices to show that ( 27 ) " ΣΕ L ei L · x q ( x ) dx C converges unconditionally to F ( q ) for each q in Ca ( C ) . For any ...
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 Haar measure 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 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 transform unique unitary vanishes vector zero