Linear Operators: Spectral operators |
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Page 1934
... seen that x * x ( § ) = 0 II.3.14 , x ( § ) = 0 and thus ܣܪ for all έ and all x * Є X * . Hence , by Corollary x - ( §I − T ) x ( § ) = 0 . -- Q.E.D. 4 THEOREM . Let T be a bounded spectral operator with resolution of the identity E ...
... seen that x * x ( § ) = 0 II.3.14 , x ( § ) = 0 and thus ܣܪ for all έ and all x * Є X * . Hence , by Corollary x - ( §I − T ) x ( § ) = 0 . -- Q.E.D. 4 THEOREM . Let T be a bounded spectral operator with resolution of the identity E ...
Page 2093
... seen that X ( F ) is a linear manifold in X , but it need not be closed even when F is closed ; an elementary counterexample is given in Colojoară and Foias [ 4 ; p.25 ] . If T is a spectral operator with resolution of the identity E ...
... seen that X ( F ) is a linear manifold in X , but it need not be closed even when F is closed ; an elementary counterexample is given in Colojoară and Foias [ 4 ; p.25 ] . If T is a spectral operator with resolution of the identity E ...
Page 2163
... seen from Corollary XV.3.7 that F ( § ) also commutes with the projec- tions in the range of E , that is , ( iii ) F ( ¿ ) E ( 0 ) = E ( 0 ) F ( § ) , ξερ ( Τ ) , for every Borel set σ . If = { σ1 , ... , σn } , π ' = { σ1 ' , ... , σ'n ...
... seen from Corollary XV.3.7 that F ( § ) also commutes with the projec- tions in the range of E , that is , ( iii ) F ( ¿ ) E ( 0 ) = E ( 0 ) F ( § ) , ξερ ( Τ ) , for every Borel set σ . If = { σ1 , ... , σn } , π ' = { σ1 ' , ... , σ'n ...
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