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 2397
PROOF . The theorem will follow as soon as it is shown that the hypotheses of
Theorem XVIII . ... Hypothesis ( ii ) has been established and is given by Lemma
7 ( iv ) . It therefore only remains to establish hypothesis ( iii ) of Theorem XVIII . 2
.
PROOF . The theorem will follow as soon as it is shown that the hypotheses of
Theorem XVIII . ... Hypothesis ( ii ) has been established and is given by Lemma
7 ( iv ) . It therefore only remains to establish hypothesis ( iii ) of Theorem XVIII . 2
.
Page 2401
Using hypothesis ( c ) , we may write this last equation as ( 5 ) q ( B - 4 ( B , A ; ) )
= ( Al ) . Now , by hypothesis , the map B > * ( B , A1 ) of A → A has norm at most
M ( M , + M ) - 1 . Thus , by Lemma VII . 3 . 4 , the equation ( 6 ) B - % ( B , Aj ) = A1
...
Using hypothesis ( c ) , we may write this last equation as ( 5 ) q ( B - 4 ( B , A ; ) )
= ( Al ) . Now , by hypothesis , the map B > * ( B , A1 ) of A → A has norm at most
M ( M , + M ) - 1 . Thus , by Lemma VII . 3 . 4 , the equation ( 6 ) B - % ( B , Aj ) = A1
...
Page 2449
On the other hand , assuming the truth of our assertion for a given n , it follows
from ( 6 ) that in + 1 Sti + 6M2t4 . Since , by hypothesis , 6M2t1 S 1 , we have In +
1 = 2t , and our assertion follows . It also follows from ( 5 ) and hypotheses ( d )
and ...
On the other hand , assuming the truth of our assertion for a given n , it follows
from ( 6 ) that in + 1 Sti + 6M2t4 . Since , by hypothesis , 6M2t1 S 1 , we have In +
1 = 2t , and our assertion follows . It also follows from ( 5 ) and hypotheses ( d )
and ...
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Contents
SPECTRAL OPERATORS | 1924 |
Introduction | 1927 |
Terminology and Preliminary Notions | 1929 |
Copyright | |
28 other sections not shown
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Common terms and phrases
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 countably additive defined Definition denote dense determined differential operator domain elements equation equivalent established exists extension fact 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 |