Linear Operators: Spectral theory |
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Page 876
Then ( y + Nie ) ( a ) = y ( 2 ) + Ni = i ( 1 + N ) , and hence 11 + N ly + Niel . Hence
( 1 + N ) < ly + Niel2 = | ( y + Nie ) ( y + Nie ) * ) = ( y + Nie ) ( y - Nie ) \ y2 + N el = \
y2I + N2 . Since this inequality must hold for all real N , a contradiction is ...
Then ( y + Nie ) ( a ) = y ( 2 ) + Ni = i ( 1 + N ) , and hence 11 + N ly + Niel . Hence
( 1 + N ) < ly + Niel2 = | ( y + Nie ) ( y + Nie ) * ) = ( y + Nie ) ( y - Nie ) \ y2 + N el = \
y2I + N2 . Since this inequality must hold for all real N , a contradiction is ...
Page 1027
Hence a belongs to the spectrum of ET . Conversely , suppose that a non - zero
scalar 2 belongs to the spectrum of ET . Then , for some non - zero æ in EH , we
have ETx = 2x . Then Tx = x + y , where y belongs to the subspace ( I - E ) H , and
...
Hence a belongs to the spectrum of ET . Conversely , suppose that a non - zero
scalar 2 belongs to the spectrum of ET . Then , for some non - zero æ in EH , we
have ETx = 2x . Then Tx = x + y , where y belongs to the subspace ( I - E ) H , and
...
Page 1227
Hence T * x = ix , or æ e D. Hence D is closed . Similarly , D is closed . Since D4
and D are clearly linear subspaces of D ( T * ) , 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 æ e D. Hence D is closed . Similarly , D is closed . Since D4
and D are clearly linear subspaces of D ( T * ) , 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
SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |
BAlgebras | 859 |
Commutative BAlgebras | 868 |
Copyright | |
57 other sections not shown
Common terms and phrases
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 Nauk neighborhood norm obtained partial positive preceding present problem projection proof properties prove range regular remark representation respectively restriction result satisfies seen sequence 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, Nelson Dunford Volume 2 of Linear Operators: General Theory. Spectral Theory, Self Adjoint Operators in Hilbert Space. Spectral Operators. I. II. III, William George Bade Volume 2 of Linear Operators: Self Adjoint Operators in Hilbert Space. Spectral Theory, Jacob T. Schwartz Volume 2; Volume 7 of Pure and applied mathematics : a series of texts and monographs Pure and applied mathematics Volume 7 of R.Courant L.Bers and J.J.Stoker Wiley classics library |
| Authors | Nelson Dunford, Jacob T. Schwartz |
| Edition | reprint |
| Publisher | Interscience Publishers, 1988 |
| Original from | Pennsylvania State University |
| Digitized | 11 Oct 2011 |
| ISBN | 0470226382, 9780470226384 |
| Length | 1070 pages |
| Export Citation | BiBTeX EndNote RefMan |