Linear Operators: Spectral theory |
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Page 876
Then ( y + Nie ) ( a ) = y ( a ) + Ni = i ( 1 + N ) , and hence 11 + N Sly + Niel .
Hence ( 1 + N ) 2 < \ y + Nie | 2 = | ( y + Nie ) ( y + Nie ) * 1 = | ( y + Nie ) ( y - Nie )
= \ y2 + Nael \ y2I + N2 . Since this inequality must hold for all real N , a
contradiction is ...
Then ( y + Nie ) ( a ) = y ( a ) + Ni = i ( 1 + N ) , and hence 11 + N Sly + Niel .
Hence ( 1 + N ) 2 < \ y + Nie | 2 = | ( y + Nie ) ( y + Nie ) * 1 = | ( y + Nie ) ( y - Nie )
= \ y2 + Nael \ y2I + N2 . Since this inequality must hold for all real N , a
contradiction is ...
Page 1027
5 shows that a is an eigenvalue and hence for some non - zero æ in H we have
Tx = ax , and hence , since T = TE , we have ( ET ) ( Ex ) = 1 Ex . Hence a belongs
to the spectrum of ET . Conversely , suppose that a non - zero scalar 1 belongs ...
5 shows that a is an eigenvalue and hence for some non - zero æ in H we have
Tx = ax , and hence , since T = TE , we have ( ET ) ( Ex ) = 1 Ex . Hence a belongs
to the spectrum of ET . Conversely , suppose that a non - zero scalar 1 belongs ...
Page 1227
Hence T * x = ix , or x € Dt . Hence Dt is closed . Similarly , D is closed . Since D
and D _ are clearly linear subspaces of D ( 7 ' * ) , it remains to show that the
spaces D ( T ) , Dr , and D are mutually orthogonal , and that their sum is D ( T * ) .
Hence T * x = ix , or x € Dt . Hence Dt is closed . Similarly , D is closed . Since D
and D _ are clearly linear subspaces of D ( 7 ' * ) , it remains to show that the
spaces D ( T ) , Dr , and D are mutually orthogonal , and that their sum is D ( T * ) .
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Contents
IX | 859 |
Eigenvalues and Eigenvectors | 903 |
Spectral Representation | 911 |
Copyright | |
15 other sections not shown
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additive Akad algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure multiplicity neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result Russian satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero
Bibliographic information
| Title | Linear Operators: Spectral theory Part 2 of Linear Operators, Jacob T. Schwartz Volume 7 of Pure and applied mathematics : a series of texts and monographs Volume 7 of Pure and applied mathematics Interscience Press Volume 7 of Pure and applied mathematics Wiley Classics Library |
| Authors | Nelson Dunford, Jacob T. Schwartz, William G. Bade, Robert G. Bartle |
| Contributor | Karreman Mathematics Research Collection |
| Publisher | Interscience Publishers, 1958 |
| Original from | the University of Michigan |
| Digitized | 2 Feb 2010 |
| ISBN | 0470226382, 9780470226384 |
| Length | 2592 pages |
| Export Citation | BiBTeX EndNote RefMan |