## Linear Operators: Spectral theory |

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Page 977

polar coordinates u = s cos q , v = s sing , we have xu + yv = rs cos ( 0 - 9 ) , and

so PR 277 o G ( u , v ) = lim i ( r ) rdre - i { rs cos ( 0 - 0 ) - n0 } dA . R→ 29 Jo By

substituting 6 ' for 6 – 9 + ( / 2 ) and simplifying , it is

) ...

polar coordinates u = s cos q , v = s sing , we have xu + yv = rs cos ( 0 - 9 ) , and

so PR 277 o G ( u , v ) = lim i ( r ) rdre - i { rs cos ( 0 - 0 ) - n0 } dA . R→ 29 Jo By

substituting 6 ' for 6 – 9 + ( / 2 ) and simplifying , it is

**seen**that G ( u , v ) = - ( - ieim) ...

Page 1024

( 1 + tr ) * ( 1 - 4 ) : Since ( 1 / N ) | tr ( B ) | < 1 and 2 # hx , the inverse operator ( I –

By ) - 1 exists and it is readily

Therefore | ( I – B ) - 11 = | ( I - BN ) - 1 ) and so ( ii ) det ( I – Bx ) | | ( I – B ) - 11 = (

det ...

( 1 + tr ) * ( 1 - 4 ) : Since ( 1 / N ) | tr ( B ) | < 1 and 2 # hx , the inverse operator ( I –

By ) - 1 exists and it is readily

**seen**that I - Box , vì = [ 1 - B = ( 1 + tr ( B ) 4 ] .Therefore | ( I – B ) - 11 = | ( I - BN ) - 1 ) and so ( ii ) det ( I – Bx ) | | ( I – B ) - 11 = (

det ...

Page 1154

Since it is clear that { ( 2 ) = Ex£ , what will be proved then , is that 2 ( 2 ) ( E ) = c (

2x2 ) ( E ) , Ee { ( 2 ) , for some constant c independent of E . This condition ( i ) ,

as is

...

Since it is clear that { ( 2 ) = Ex£ , what will be proved then , is that 2 ( 2 ) ( E ) = c (

2x2 ) ( E ) , Ee { ( 2 ) , for some constant c independent of E . This condition ( i ) ,

as is

**seen**from Corollary III . 11 . 6 , is a consequence of the assertion that ( ii ) 2...

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### Contents

IX | 859 |

extensive presentation of applications of the spectral theorem | 911 |

Miscellaneous Applications | 937 |

Copyright | |

20 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

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additive adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f give given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero