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 1934
PROOF . Using Theorem 2 , we see that if g ( x ) is void , then x ( ) is everywhere
defined , single valued , and hence entire . Since , by VII . 3 . 4 , lim x * x ( ) = lim x
* R ( É ; T ' ) . x = 0 , $ 200 $ 700 it is seen that x * x ( $ ) = 0 for all & and all x * e ...
PROOF . Using Theorem 2 , we see that if g ( x ) is void , then x ( ) is everywhere
defined , single valued , and hence entire . Since , by VII . 3 . 4 , lim x * x ( ) = lim x
* R ( É ; T ' ) . x = 0 , $ 200 $ 700 it is seen that x * x ( $ ) = 0 for all & and all x * e ...
Page 2163
8 , it is seen that , for some constant K , 3 F46 . JE ( 0 . ) – É PLE ) E ( 0 ) | FIĚ Ž { P
( 6 . ) – F ( E ) B ( 0 , 05 ) SK sup | F ( Xi ) – F ( 81 ) , where the supremum is taken
over those i and j for which o ; o ; is not void . If by the norm ( 7 | is understood ...
8 , it is seen that , for some constant K , 3 F46 . JE ( 0 . ) – É PLE ) E ( 0 ) | FIĚ Ž { P
( 6 . ) – F ( E ) B ( 0 , 05 ) SK sup | F ( Xi ) – F ( 81 ) , where the supremum is taken
over those i and j for which o ; o ; is not void . If by the norm ( 7 | is understood ...
Page 2185
Since every function f in C ( 1 ) is bounded and Borel measurable , the integral s f
( a ) A ( dà ) exists and it is seen from ( i ) that ( ii ) S * ( 4 ( da ) , feC ( 1 ) . It will
now be shown that A is a spectral measure in X * . By placing f = 1 in ( ii ) , it is ...
Since every function f in C ( 1 ) is bounded and Borel measurable , the integral s f
( a ) A ( dà ) exists and it is seen from ( i ) that ( ii ) S * ( 4 ( da ) , feC ( 1 ) . It will
now be shown that A is a spectral measure in X * . By placing f = 1 in ( ii ) , it is ...
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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 |