Linear Operators: Spectral operators |
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Page 2152
... prove that M ( S ) is closed . In other words , we may and shall assume that 8 is the complement of an open subinterval y of Io . Let { x } be a sequence in X , convergent to the point x , and with p ( x ) = y . To prove ( C ) it will ...
... prove that M ( S ) is closed . In other words , we may and shall assume that 8 is the complement of an open subinterval y of Io . Let { x } be a sequence in X , convergent to the point x , and with p ( x ) = y . To prove ( C ) it will ...
Page 2236
... proves the first part of ( v ) . To prove the second part of ( v ) it is sufficient , since ƒ ( T ) is closed , to show that f ( T ) is everywhere defined ( cf. the closed graph theorem ( II.2.4 ) ) . By ( ii ) , D ( f ( T ) ) ≥ E ( e ) ...
... proves the first part of ( v ) . To prove the second part of ( v ) it is sufficient , since ƒ ( T ) is closed , to show that f ( T ) is everywhere defined ( cf. the closed graph theorem ( II.2.4 ) ) . By ( ii ) , D ( f ( T ) ) ≥ E ( e ) ...
Page 2459
... prove this , we reason as follows . We may write Hx Y1 + y2 , where y1 € Σac ( H ) , Y2 € Hac ( H ) . We have ( I + H2 ) − 1 — ( ¿ I — H ) − 1 ( ¿ I — H ) 1 , Theorem XII.2.6 , and therefore , using what we have already proved , we ...
... prove this , we reason as follows . We may write Hx Y1 + y2 , where y1 € Σac ( H ) , Y2 € Hac ( H ) . We have ( I + H2 ) − 1 — ( ¿ I — H ) − 1 ( ¿ I — H ) 1 , Theorem XII.2.6 , and therefore , using what we have already proved , we ...
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