Linear Operators, Part 2 |
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Page 2391
... lemma . Q.E.D. + 5 COROLLARY . Let the hypotheses of Lemma 4 be satisfied and let 0 < λ2 < ∞ . Suppose in the notation of Lemma 4 that A * ( λ1 ) # 0 , A ̄ ( 1 ) 0. Then for λ = λ , lying on any sufficiently short transversal to the ...
... lemma . Q.E.D. + 5 COROLLARY . Let the hypotheses of Lemma 4 be satisfied and let 0 < λ2 < ∞ . Suppose in the notation of Lemma 4 that A * ( λ1 ) # 0 , A ̄ ( 1 ) 0. Then for λ = λ , lying on any sufficiently short transversal to the ...
Page 2395
... lemma . Q.E.D. 9 COROLLARY . Let the hypotheses of Lemma 7 be satisfied , and in particular let A * ( λ ) and A− ( λ \ ) 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 * ( λ ) and A− ( λ \ ) 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 f ( t ) = 0 , uniformly for 0≤t < ∞o . 00 + 1111 μερ + Hence , by formula ( 24 ) of the proof of Lemma 3 , Ĵu ( t ) ~ e - itu ; gu ( t ) = ~ -iμe - itufu ( t ) + ettuf ...
... Lemma 1 ( cf. the para- graph following formula ( 14 ) ) that lim f ( t ) = 0 , uniformly for 0≤t < ∞o . 00 + 1111 μερ + Hence , by formula ( 24 ) of the proof of Lemma 3 , Ĵu ( t ) ~ e - itu ; gu ( t ) = ~ -iμe - itufu ( t ) + ettuf ...
Contents
SPECTRAL OPERATORS | 1924 |
The Canonical Reduction of a Spectral Operator | 1939 |
Bounded Spectral Operators in Hilbert Space | 1947 |
Copyright | |
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A₁ adjoint operator algebra of projections Amer analytic arbitrary B-algebra B*-algebra B₁ Banach space Boolean algebra Borel sets boundary conditions bounded Borel function bounded linear operator bounded operator closed operator commuting compact complex numbers complex plane converges Corollary countably additive Definition denote dense differential operator Doklady Akad domain eigenvalues elements equation exists finite number follows from Lemma formal differential operator formula function f H₁ H₂ Hence Hilbert space hypothesis identity inequality integral invariant inverse L₁ Lebesgue Lemma Math multiplicity Nauk SSSR norm operators in Hilbert perturbation polynomial PROOF properties prove quasi-nilpotent resolution Russian S₁ satisfies scalar type operator scalar type spectral Section sequence shows spectral measure spectral operator spectral theory spectrum subset subspace Suppose trace class type spectral operator unbounded uniformly bounded unique vector zero