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
Since , by VII.3.4 , lim x * x ( ) = lim x * R ( E ; T ' ) . x = 0 , $ x $ > $ → it is seen that x * x ( $ ) = 0 for all & and all x * e X * . Hence , by Corollary II.3.14 , x ( 8 ) = 0 and thus x = ( $ I - T ' ) 3 ( E ) = 0 .
Since , by VII.3.4 , lim x * x ( ) = lim x * R ( E ; T ' ) . x = 0 , $ x $ > $ → it is seen that x * x ( $ ) = 0 for all & and all x * e X * . Hence , by Corollary II.3.14 , x ( 8 ) = 0 and thus x = ( $ I - T ' ) 3 ( E ) = 0 .
Page 2163
It is seen from Corollary XV.3.7 that F ( $ ) also commutes with the projections in the range of E , that is , ( iii ) F ( $ ) E ( 0 ) = E ( 0 ) F ...
It is seen from Corollary XV.3.7 that F ( $ ) also commutes with the projections in the range of E , that is , ( iii ) F ( $ ) E ( 0 ) = E ( 0 ) F ...
Page 2183
On the other hand , since PB contains all the elements of A ( B ) which have the form ( ii ) , and since this set of elements is dense in A ( B ) , it will be seen that PA ( T ) = A ( B ) . Moreover , it is clear that ( I – P ) B = R ...
On the other hand , since PB contains all the elements of A ( B ) which have the form ( ii ) , and since this set of elements is dense in A ( B ) , it will be seen that PA ( T ) = A ( B ) . Moreover , it is clear that ( I – P ) B = R ...
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Contents
SPECTRAL OPERATORS | 1924 |
Introduction | 1927 |
Terminology and Preliminary Notions | 1929 |
Copyright | |
47 other sections not shown
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adjoint operator Amer analytic apply arbitrary assumed B-space Banach space belongs Boolean algebra Borel set boundary conditions bounded bounded operator Chapter clear closed commuting compact complex constant contains continuous converges Corollary corresponding defined Definition denote dense determined differential operator domain elements equation equivalent established exists extension finite follows formal formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear operator Math Moreover multiplicity norm perturbation plane positive preceding present problem projections PROOF properties prove range resolution resolvent restriction Russian satisfies scalar type seen sequence shown shows similar solution spectral measure spectral operator spectrum subset sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector 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 |