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 2152
Spectral Theory : Self Adjoint Operators in Hilbert Space Nelson Dunford, Jacob
Theodore Schwartz. To prove ( C ) , let d be a closed subset of the complex plane
and let M ( S ) = { 2 x € X , ( x ) Ş8 } . It will be shown that M ( 8 ) is closed .
Spectral Theory : Self Adjoint Operators in Hilbert Space Nelson Dunford, Jacob
Theodore Schwartz. To prove ( C ) , let d be a closed subset of the complex plane
and let M ( S ) = { 2 x € X , ( x ) Ş8 } . It will be shown that M ( 8 ) is closed .
Page 2236
To prove ( vi ) , let x be in D ( f ( T ) + g ( T ) ) and let { en } be as above . Then ,
since T | Elen ) X is bounded , statements ( i ) and ( ii ) and the functional calculus
of bounded operators ( cf . VII . 3 . 10 ) may be applied to conclude that lim ( f + g )
...
To prove ( vi ) , let x be in D ( f ( T ) + g ( T ) ) and let { en } be as above . Then ,
since T | Elen ) X is bounded , statements ( i ) and ( ii ) and the functional calculus
of bounded operators ( cf . VII . 3 . 10 ) may be applied to conclude that lim ( f + g )
...
Page 2459
If xn e Lac ( H ) and limn + Xn = x , then , by what we have already proved , we
may write x = y1 + y2 + Ys , where yı e Lac ( H ) and Y2 , Y3 are orthogonal to Lac
( H ) ... Using this last fact , it is easy to prove assertion ( c ) of the present lemma .
If xn e Lac ( H ) and limn + Xn = x , then , by what we have already proved , we
may write x = y1 + y2 + Ys , where yı e Lac ( H ) and Y2 , Y3 are orthogonal to Lac
( H ) ... Using this last fact , it is easy to prove assertion ( c ) of the present lemma .
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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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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 |