Linear Operators: Part III: Spectral Operators [by] Nelson Dunford and Jacob T. Schwartz, with the Assistance of William G. Bade and Robert G. Bartle, Volume 1 |
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Page 2391
This shows that f = R ( A ) ( AI – T ) f for each fe D ( T ) . Thus , if le R , Al > ? , and
A ( X ) + 0 , then ( XI – T ) - 1 = R ( 1 ; T ) exists and equals R ( A ) , completing the
proof of the present lemma . Q . E . D . 5 COROLLARY . Let the hypotheses of ...
This shows that f = R ( A ) ( AI – T ) f for each fe D ( T ) . Thus , if le R , Al > ? , and
A ( X ) + 0 , then ( XI – T ) - 1 = R ( 1 ; T ) exists and equals R ( A ) , completing the
proof of the present lemma . Q . E . D . 5 COROLLARY . Let the hypotheses of ...
Page 2396
It follows from this formula just as in the proof of Lemma 1 ( cf . the paragraph
following formula ( 14 ) ) that lim 1fu ( t ) = 0 , uniformly for Ost < 0 . HEP + Hence ,
by formula ( 24 ) of the proof of Lemma 3 , ĝu ( t ) ~ e - ttu ; Ô ( t ) = - ime - itufu ( t )
+ ...
It follows from this formula just as in the proof of Lemma 1 ( cf . the paragraph
following formula ( 14 ) ) that lim 1fu ( t ) = 0 , uniformly for Ost < 0 . HEP + Hence ,
by formula ( 24 ) of the proof of Lemma 3 , ĝu ( t ) ~ e - ttu ; Ô ( t ) = - ime - itufu ( t )
+ ...
Page 2479
regarded as a subspace of the larger space H ' of Lemma 15 , while equally
plainly H , may be regarded as the restriction to H , of the operator H of Lemma
15 ( cf . ( 33 ) – ( 36 ) above ) . Let Q be the projection of H ' onto its subspace Hı ,
and ...
regarded as a subspace of the larger space H ' of Lemma 15 , while equally
plainly H , may be regarded as the restriction to H , of the operator H of Lemma
15 ( cf . ( 33 ) – ( 36 ) above ) . Let Q be the projection of H ' onto its subspace Hı ,
and ...
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Contents
SPECTRAL OPERATORS XV Spectral Operators | 1924 |
Introduction | 1925 |
Terminology and Preliminary Notions | 1928 |
Copyright | |
32 other sections not shown
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analytic apply arbitrary assumed B-space Banach space Boolean algebra Borel sets boundary conditions bounded bounded Borel bounded operator Chapter clear clearly closure commuting compact complex consider constant contained converges Corollary corresponding countably additive defined Definition denote dense determined differential operator domain elements equation equivalent established example exists extension fact finite follows formal formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator manifold Math Moreover multiplicity norm positive preceding present problem projections PROOF properties prove range regular resolution resolvent respectively restriction Russian satisfies scalar type seen sequence shown shows spectral measure spectral operator spectrum statement strongly subset subspace sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector weakly zero
Bibliographic information
| Title | Linear Operators: Part III: Spectral Operators [by] Nelson Dunford and Jacob T. Schwartz, with the Assistance of William G. Bade and Robert G. Bartle, Volume 1 Pure and applied mathematics, v. 7 |
| Authors | Nelson Dunford, Jacob T. Schwartz, William G. Bade, Robert G. Bartle |
| Editors | L. Bers, J. J. Stoker |
| Publisher | Wiley-Interscience, 1957 |
| Original from | the University of Michigan |
| Digitized | 2 Feb 2010 |
| ISBN | 0471226394, 9780471226390 |
| Length | 667 pages |
| Export Citation | BiBTeX EndNote RefMan |