Linear Operators, Part 2 |
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Page 2032
... following lemma will be needed . 14 LEMMA . If both T and T are functions for some T in T ( RN ) and some a , then so are T and T functions , and T ( s ) = T ( s ) , ( T ) ( s ) = ( T ) ( 8 ) . · PROOF . The statement about T is already ...
... following lemma will be needed . 14 LEMMA . If both T and T are functions for some T in T ( RN ) and some a , then so are T and T functions , and T ( s ) = T ( s ) , ( T ) ( s ) = ( T ) ( 8 ) . · PROOF . The statement about T is already ...
Page 2395
... preceding lemma . Let A ( λ ) = A ( σ1 ( · , μ ( λ ) ) ) , B ( λ ) = A ( 03 ( ⋅ , μ ( λ ) ) ) and represent R ( λ ; T ) as the integral operator whose kernel is given by Corollary 6 , using σ3 for the function o of that lemma . Then A ...
... preceding lemma . Let A ( λ ) = A ( σ1 ( · , μ ( λ ) ) ) , B ( λ ) = A ( 03 ( ⋅ , μ ( λ ) ) ) and represent R ( λ ; T ) as the integral operator whose kernel is given by Corollary 6 , using σ3 for the function o of that lemma . Then 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