Linear Operators: Spectral theory |
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Page 893
In summary we state the following theorem . 1 THEOREM . Let E be a bounded
self adjoint spectral measure in Hilbert space defined on a field of subsets of a
set S . Then the map | → T ( 1 ) defined by the equation T ( ) = | - | ( 8 ) E ( ds ) , te
B ...
In summary we state the following theorem . 1 THEOREM . Let E be a bounded
self adjoint spectral measure in Hilbert space defined on a field of subsets of a
set S . Then the map | → T ( 1 ) defined by the equation T ( ) = | - | ( 8 ) E ( ds ) , te
B ...
Page 900
and thus there is a bounded function to on S with f ( s ) = to ( s ) except for s in a
set having E measure zero . If f is E - measurable then fo is a bounded E -
measurable function , i . e . , an element of the B * - algebra B ( S , E ) . The
algebra EB ...
and thus there is a bounded function to on S with f ( s ) = to ( s ) except for s in a
set having E measure zero . If f is E - measurable then fo is a bounded E -
measurable function , i . e . , an element of the B * - algebra B ( S , E ) . The
algebra EB ...
Page 1452
Suppose that such a u exists . Then , by Theorem XII . 2 . 6 , ( Tx , x ) = E ( dx ) x2
= ulx | ? , 2 e ( T ) , so that if \ x is bounded , ( Tx , x ) is bounded below .
Conversely , suppose that for each n , en = ( - 0 , - n ) n o ( T ) is non - void . By
Theorem ...
Suppose that such a u exists . Then , by Theorem XII . 2 . 6 , ( Tx , x ) = E ( dx ) x2
= ulx | ? , 2 e ( T ) , so that if \ x is bounded , ( Tx , x ) is bounded below .
Conversely , suppose that for each n , en = ( - 0 , - n ) n o ( T ) is non - void . By
Theorem ...
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Contents
IX | 859 |
Eigenvalues and Eigenvectors | 903 |
Spectral Representation | 911 |
Copyright | |
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