Linear Operators: Spectral Theory : Self Adjoint Operators in Hilbert Space, Volume 2 |
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Page 2174
In a Hilbert space the condition that lettr | SM for all te R implies that T is
equivalent to a self adjoint operator and hence is a scalar type operator with real
spectrum . ( This follows from Lemma XV . 6 . 1 which implies that the bounded
group G ...
In a Hilbert space the condition that lettr | SM for all te R implies that T is
equivalent to a self adjoint operator and hence is a scalar type operator with real
spectrum . ( This follows from Lemma XV . 6 . 1 which implies that the bounded
group G ...
Page 2308
17 and the remark preceding Definition XIII . 2 . 29 , that S is closed . From this , it
follows immediately that S - XI is closed , and hence from Lemma XII . 1 . 5 , that (
8 – 17 ) - 1 is closed . Since this latter operator is everywhere defined , it follows ...
17 and the remark preceding Definition XIII . 2 . 29 , that S is closed . From this , it
follows immediately that S - XI is closed , and hence from Lemma XII . 1 . 5 , that (
8 – 17 ) - 1 is closed . Since this latter operator is everywhere defined , it follows ...
Page 2357
then it is clear that L is a bounded operator and that ( S – XI ) - = ( T - XI ) - ' L .
Hence ( P + N ) ( S – XI ) - v = P ( S – XI ) - ° + N ( S — 21 ) - " = P ( T – XI ) - L + N
( S – XI ) - v is a bounded operator which is compact if P ( T – XI ) - " is compact (
cf .
then it is clear that L is a bounded operator and that ( S – XI ) - = ( T - XI ) - ' L .
Hence ( P + N ) ( S – XI ) - v = P ( S – XI ) - ° + N ( S — 21 ) - " = P ( T – XI ) - L + N
( S – XI ) - v is a bounded operator which is compact if P ( T – XI ) - " is compact (
cf .
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Bibliographic information
| Title | Linear Operators: Spectral Theory : Self Adjoint Operators in Hilbert Space, Volume 2 Volume 7 of Pure and applied mathematics |
| Authors | Nelson Dunford, Jacob Theodore Schwartz |
| Publisher | Wiley Interscience, 1963 |
| Original from | the University of Michigan |
| Digitized | 2 Feb 2010 |
| ISBN | 0471226394, 9780471226390 |
| Length | 2592 pages |
| Export Citation | BiBTeX EndNote RefMan |