## Linear Operators: Spectral theory |

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Page 898

If we put E ( d ) = 0 when d n o ( T ) is void , then

from Theorem 1 and

defined spectral measure associated , in

...

If we put E ( d ) = 0 when d n o ( T ) is void , then

**Corollary**4 follows immediatelyfrom Theorem 1 and

**Corollary**IX . 3 . 15 . Q . E . D . 5 DEFINITIon . The uniquelydefined spectral measure associated , in

**Corollary**4 , with the normal operator T...

Page 1301

Proceeding inductively we see that vck ) ( b ) = 0 , 0 Sk 2n - 1 . However , as vo

satisfies an equation of order 2n , v , must be identically zero . This contradiction

completes the proof . Q . E . D . 23

Proceeding inductively we see that vck ) ( b ) = 0 , 0 Sk 2n - 1 . However , as vo

satisfies an equation of order 2n , v , must be identically zero . This contradiction

completes the proof . Q . E . D . 23

**COROLLARY**. Let t be a formal differential ...Page 1710

It follows from

2 ) on a suitable Lebesgue null set for each 2 in N , we obtain a function W ; such

that W ; ( : , a ) is in Co ( I ) for all a , and such that TW : ( : , 2 ) = 2W : ( : , 2 ) , tel ...

It follows from

**Corollary**4 that if we put W ; ( : , 2 ) = 0 for a € N and modify Wi ( t ,2 ) on a suitable Lebesgue null set for each 2 in N , we obtain a function W ; such

that W ; ( : , a ) is in Co ( I ) for all a , and such that TW : ( : , 2 ) = 2W : ( : , 2 ) , tel ...

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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 |

### Common terms and phrases

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