Linear Operators: Part III: Spectral Operators [by] Nelson Dunford and Jacob T. Schwartz, with the Assistance of William G. Bade and Robert G. Bartle, Volume 1 |
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Page 2391
... lemma . Q.E.D. + 5 COROLLARY . Let the hypotheses of Lemma 4 be satisfied and let 0 < λ1 < ∞ . Suppose in the notation of Lemma 4 that A * ( \ 1 ) ‡ 0 , A ̄ ( λ1 ) ‡ 0 . Then for λ = λ1 lying on any sufficiently short transversal to ...
... lemma . Q.E.D. + 5 COROLLARY . Let the hypotheses of Lemma 4 be satisfied and let 0 < λ1 < ∞ . Suppose in the notation of Lemma 4 that A * ( \ 1 ) ‡ 0 , A ̄ ( λ1 ) ‡ 0 . Then for λ = λ1 lying on any sufficiently short transversal to ...
Page 2395
... lemma . Q.E.D. 9 COROLLARY . Let the hypotheses of Lemma 7 be satisfied , and in particular let A * ( \ ) and A ̄ ( \ ) be non - vanishing for 0 ≤ λ < ∞ . Let f and g be a pair of functions in C [ 0 , ∞ ) which vanish outside a ...
... lemma . Q.E.D. 9 COROLLARY . Let the hypotheses of Lemma 7 be satisfied , and in particular let A * ( \ ) and A ̄ ( \ ) be non - vanishing for 0 ≤ λ < ∞ . Let f and g be a pair of functions in C [ 0 , ∞ ) which vanish outside a ...
Page 2396
... Lemma 1 ( cf. the para- graph following formula ( 14 ) ) that lim 141 → 00 HEP + f ( t ) = 0 , uniformly for 0≤t < ∞ . Hence , by formula ( 24 ) of the proof of Lemma 3 , Î1 ( t ) ~ e - itu ; gu ( t ) = -iμe - stufu ( t ) + e1tuf ...
... Lemma 1 ( cf. the para- graph following formula ( 14 ) ) that lim 141 → 00 HEP + f ( t ) = 0 , uniformly for 0≤t < ∞ . Hence , by formula ( 24 ) of the proof of Lemma 3 , Î1 ( t ) ~ e - itu ; gu ( t ) = -iμe - stufu ( t ) + e1tuf ...
Contents
SPECTRAL OPERATORS | 1924 |
An Operational Calculus for Bounded Spectral | 1941 |
Bounded Spectral Operators in Hilbert Space | 1947 |
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A₁ Acad adjoint operator algebra of projections Amer analytic arbitrary B-algebra B-space B*-algebra B₁ Banach space Boolean algebra Borel sets boundary conditions bounded linear operator bounded operator closed operator commuting compact complex numbers complex plane converges Corollary countably additive Definition dense differential operator Dokl Doklady Akad eigenvalues elements equation exists formal differential operator formula function f H₁ H₂ Hence Hilbert space hypothesis ibid identity inequality inverse Krein L₁ Lemma locally convex spaces multiplicity Nauk SSSR norm normal operators operators in Hilbert perturbation polynomial Proc PROOF properties prove Pure Appl quasi-nilpotent resolution Russian S₁ satisfies scalar operator scalar type operator scalar type spectral Section semi-group sequence shows spectral measure spectral operator spectral theory spectrum subset Suppose topology trace class type spectral operator unbounded uniformly bounded vector zero