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 2154
Since the manifolds { x | ( T – XI ) mx = 0 } increase with m , it follows that ( T – XI )
n + 1X + { x | ( T - XI ) n + 1x = 0 } is dense in X . By induction , it is seen that the
manifold ( T – 17 ) n + kX + { x | ( T – 27 ) n + Kx = 0 } is dense in X for all k 20 .
Since the manifolds { x | ( T – XI ) mx = 0 } increase with m , it follows that ( T – XI )
n + 1X + { x | ( T - XI ) n + 1x = 0 } is dense in X . By induction , it is seen that the
manifold ( T – 17 ) n + kX + { x | ( T – 27 ) n + Kx = 0 } is dense in X for all k 20 .
Page 2214
To prove the converse , let A leave invariant every closed linear manifold which is
invariant under every element of B and let B , be the strong closure of B . Then it
is clear that A leaves invariant every closed linear manifold which is invariant ...
To prove the converse , let A leave invariant every closed linear manifold which is
invariant under every element of B and let B , be the strong closure of B . Then it
is clear that A leaves invariant every closed linear manifold which is invariant ...
Page 2270
In particular the manifolds E * * * , E * € B * , are X - closed . PROOF . That E * X *
= { x * | ( I – E ) * 2 * = 0 } is X - closed is a standard elementary property of the
null manifold of an adjoint operator . Let { E , } be an arbitrary set ( sequence ) of ...
In particular the manifolds E * * * , E * € B * , are X - closed . PROOF . That E * X *
= { x * | ( I – E ) * 2 * = 0 } is X - closed is a standard elementary property of the
null manifold of an adjoint operator . Let { E , } be an arbitrary set ( sequence ) of ...
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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 |