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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10 ) that T ( fxe ) X = T ( fxe ) E ( ē ) x = T ( fXeě ) x = T ( fxë ) E ( e ) x = T ' ( fXē ) x ,
so that Qo is well defined on Ueeso E ( e ) X . It thus follows from Lemma 6 that T
( f ) is a closed , densely defined operator . Moreover , statement ( g ) follows ...
10 ) that T ( fxe ) X = T ( fxe ) E ( ē ) x = T ( fXeě ) x = T ( fxë ) E ( e ) x = T ' ( fXē ) x ,
so that Qo is well defined on Ueeso E ( e ) X . It thus follows from Lemma 6 that T
( f ) is a closed , densely defined operator . Moreover , statement ( g ) follows ...
Page 2246
it follows that R ( 1 ) is a bounded operator whose range is contained in the
domain of C . It is clear then that ( XI – C ) R ( a ) = x for x in H and R ( a ) ( 21 – C
) x = x for x in D ( C ) , so that R ( a ) = R ( a ; C ) and 1 € ( C ) . On the other hand ,
if X ...
it follows that R ( 1 ) is a bounded operator whose range is contained in the
domain of C . It is clear then that ( XI – C ) R ( a ) = x for x in H and R ( a ) ( 21 – C
) x = x for x in D ( C ) , so that R ( a ) = R ( a ; C ) and 1 € ( C ) . On the other hand ,
if X ...
Page 2459
Statement ( b ) of our lemma follows at once . If xn Lac ( H ) ... Thus statement ( a )
of our theorem follows . If F is a bounded ... If ( il – H ) - 1 [ ac ( H ) were not dense
in Lac ( H ) , it would follow by the Hahn - Banach theorem ( cf . Corollary II .
Statement ( b ) of our lemma follows at once . If xn Lac ( H ) ... Thus statement ( a )
of our theorem follows . If F is a bounded ... If ( il – H ) - 1 [ ac ( H ) were not dense
in Lac ( H ) , it would follow by the Hahn - Banach theorem ( cf . Corollary II .
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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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Common terms and phrases
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 |