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 1953
If E ( { } ) = 0 , then ( AI – T ' ) X is dense in X. PROOF . First suppose that = 0.If TX
is not dense then , by Corollary II.3.13 , there is an x * in X * with x * # 0 and ** TX
= 0 . Let x1 # 0 and define the operator A by the equation Ax = x * ( x ) x1 , so ...
If E ( { } ) = 0 , then ( AI – T ' ) X is dense in X. PROOF . First suppose that = 0.If TX
is not dense then , by Corollary II.3.13 , there is an x * in X * with x * # 0 and ** TX
= 0 . Let x1 # 0 and define the operator A by the equation Ax = x * ( x ) x1 , so ...
Page 2137
Even though ( A ) is taken as a standing assumption throughout this section , it
will be indicated parenthetically in the statement of each lemma in the proof of
which it is used . 1 LEMMA ( A ) . If a , ß are complex numbers and x , y are
vectors in ...
Even though ( A ) is taken as a standing assumption throughout this section , it
will be indicated parenthetically in the statement of each lemma in the proof of
which it is used . 1 LEMMA ( A ) . If a , ß are complex numbers and x , y are
vectors in ...
Page 2192
PROOF . This follows from Corollary 12 and Lemma 1 . Q.E.D. 14 COROLLARY .
Let X be a weakly complete B - space . Let A ( T ) be the full algebra generated by
a family 7 of commuting spectral operators . If the Boolean algebra B ...
PROOF . This follows from Corollary 12 and Lemma 1 . Q.E.D. 14 COROLLARY .
Let X be a weakly complete B - space . Let A ( T ) be the full algebra generated by
a family 7 of commuting spectral operators . If the Boolean algebra B ...
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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 |