Linear Operators, Part 2 |
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Page 2195
... complete ( o - complete ) as an abstract Boolean algebra if each subset ( sequence ) of B has a greatest lower bound and a least upper bound in B. The Boolean algebra B is said to be complete ( o - complete ) if it is complete ( o ...
... complete ( o - complete ) as an abstract Boolean algebra if each subset ( sequence ) of B has a greatest lower bound and a least upper bound in B. The Boolean algebra B is said to be complete ( o - complete ) if it is complete ( o ...
Page 2214
... complete space . Then an operator is in the weakly closed algebra generated by B if and only if it leaves invariant every closed linear manifold which is invariant under every member of B. PROOF . It was observed in the preceding proof ...
... complete space . Then an operator is in the weakly closed algebra generated by B if and only if it leaves invariant every closed linear manifold which is invariant under every member of B. PROOF . It was observed in the preceding proof ...
Page 2217
... complete . By Lemma 4 there is a monotone increasing generalized sequence { E } in B1 and an x in X such that the ... complete . Thus , by Corollary 7 , B ( x ) is not complete . On the other hand , B ( x ) is , according to Corollary 11 ...
... complete . By Lemma 4 there is a monotone increasing generalized sequence { E } in B1 and an x in X such that the ... complete . Thus , by Corollary 7 , B ( x ) is not complete . On the other hand , B ( x ) is , according to Corollary 11 ...
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