Linear Operators, Part 2 |
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Page 1953
... PROOF . The proof will be divided into two cases depending on whether the projection E ( { 0 } ) = 0 or not . First suppose that E ( { 0 } ) = 0 . Then since the range of T is closed , it follows from Corollary 12 that TX = X. By Lemma ...
... PROOF . The proof will be divided into two cases depending on whether the projection E ( { 0 } ) = 0 or not . First suppose that E ( { 0 } ) = 0 . Then since the range of T is closed , it follows from Corollary 12 that TX = X. By Lemma ...
Page 2137
... PROOF . The proof of Corollary XV.3.3 may be used to prove the present lemma . Q.E.D. 3 LEMMA ( A ) . Let o be a set of complex numbers , and o ' its com- plement . If x + y = x1 + y1 , where σ ( x ) , o ( x1 ) ≤ o and o ( y ) , o ( y1 ) ...
... PROOF . The proof of Corollary XV.3.3 may be used to prove the present lemma . Q.E.D. 3 LEMMA ( A ) . Let o be a set of complex numbers , and o ' its com- plement . If x + y = x1 + y1 , where σ ( x ) , o ( x1 ) ≤ o and o ( y ) , o ( y1 ) ...
Page 2192
... proof of the lemma . E ( 8 ) = 1 Q.E.D. 12 COROLLARY . Let A be an algebra of operators in a weakly complete B - space X. Suppose that A is topologically and algebraically isomorphic to some B - algebra of bounded continuous functions ...
... proof of the lemma . E ( 8 ) = 1 Q.E.D. 12 COROLLARY . Let A be an algebra of operators in a weakly complete B - space X. Suppose that A is topologically and algebraically isomorphic to some B - algebra of bounded continuous functions ...
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