Linear Operators: Spectral operators |
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Page 2194
... closure in the weak as in the strong operator topology . Thus the strong and weak operator closures of an algebra of ... closure of the algebra generated by the projections in its resolution of the identity , the problem of the present ...
... closure in the weak as in the strong operator topology . Thus the strong and weak operator closures of an algebra of ... closure of the algebra generated by the projections in its resolution of the identity , the problem of the present ...
Page 2217
... closure . By Lemma 3 , B is bounded and thus B1 is also a bounded Boolean algebra of projections in X. Suppose that B is not complete . By Lemma 4 there is a monotone increasing generalized sequence { E } in B1 and an x in X such that ...
... closure . By Lemma 3 , B is bounded and thus B1 is also a bounded Boolean algebra of projections in X. Suppose that B is not complete . By Lemma 4 there is a monotone increasing generalized sequence { E } in B1 and an x in X such that ...
Page 2234
... closure contained in U , it may be supposed , since ƒ ( T ) is independent of { e } , that e = e1 . If x is in E ( e ) X , then , by the paragraph preceding Definition 8 , f ( T | E ( en ) X ) x = ƒ ( T | E ( e ) X ) x for n ≥1 so that ...
... closure contained in U , it may be supposed , since ƒ ( T ) is independent of { e } , that e = e1 . If x is in E ( e ) X , then , by the paragraph preceding Definition 8 , f ( T | E ( en ) X ) x = ƒ ( T | E ( e ) X ) x for n ≥1 so that ...
Contents
SPECTRAL OPERATORS | 1924 |
Spectral Operators | 1925 |
Terminology and Preliminary Notions | 1928 |
Copyright | |
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A₁ algebra Amer analytic applications arbitrary B-space Banach Banach space Boolean algebra Borel sets boundary bounded Chapter clear closed commuting compact complex constant contains continuous converges Corollary corresponding defined Definition denote dense determined differential operator Doklady Akad elements equation equivalent established example exists extension finite follows formula function given gives H₁ Hence Hilbert space hypothesis identity integral invariant inverse Lemma limit linear operators Math multiplicity Nauk SSSR norm normal perturbation plane positive preceding present problem Proc projections PROOF properties prove range resolution resolvent restriction Russian satisfies scalar type sequence shown shows similar spectral measure spectral operator spectrum subset subspace sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector zero