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 2149
Condition ( A ) of Section 2 is satisfied if the spectrum of T is nowhere dense in
the complex plane . PROOF . If the resolvent set is dense , then any two analytic ,
or even continuous , extensions of R ( a ; T ' ) x must coincide on their common ...
Condition ( A ) of Section 2 is satisfied if the spectrum of T is nowhere dense in
the complex plane . PROOF . If the resolvent set is dense , then any two analytic ,
or even continuous , extensions of R ( a ; T ' ) x must coincide on their common ...
Page 2156
are both dense in X . Since M , is dense in X , the manifold ( 1 , 1 – T ) \ M , + { x | (
9 , 1 – T ' ) Nx = 0 } is dense in X , so that ( 9 , 1 – T ) * ( 121 – T ) NX + { x | ( 9 , 1 –
T ) ^ x = 0 } + { x | ( 1 , 1 – T ) x = 0 } is also dense in X . By Lemma 7 , 0 ( 2 ) cy ...
are both dense in X . Since M , is dense in X , the manifold ( 1 , 1 – T ) \ M , + { x | (
9 , 1 – T ' ) Nx = 0 } is dense in X , so that ( 9 , 1 – T ) * ( 121 – T ) NX + { x | ( 9 , 1 –
T ) ^ x = 0 } + { x | ( 1 , 1 – T ) x = 0 } is also dense in X . By Lemma 7 , 0 ( 2 ) cy ...
Page 2159
The union of all intervals of constancy relative to T is an open set dense in lo .
Proof . It is clear that the union of intervals of constancy is open . To see that it is
dense , let y be a closed subarc of T , having positive length and let Yn = { 10 do
ey ...
The union of all intervals of constancy relative to T is an open set dense in lo .
Proof . It is clear that the union of intervals of constancy is open . To see that it is
dense , let y be a closed subarc of T , having positive length and let Yn = { 10 do
ey ...
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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 |