## Linear Operators: Spectral theory |

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Page 861

If To

Tile , then az = To'z for every z e X. Also xa = Tra = e = 7 ; ' ( Te ) = T2 ( ex ) = ( To'

e ) x Thus x - 1

...

If To

**exists**in B ( X ) , then Te [ ( T = ' y ) x ] = yz , ( T ' y ) 2 = T ; ' ( yz ) , and if a =Tile , then az = To'z for every z e X. Also xa = Tra = e = 7 ; ' ( Te ) = T2 ( ex ) = ( To'

e ) x Thus x - 1

**exists**and T7 % = x - 1z . = ax . 2 DEFINITION . An element æ in a...

Page 1057

En lyn provided only that the limit in the braces in this last equation

to complete the proof of the present lemma , it suffices to show that 2 ( y ) 2 ( y ) (

3 ) 0 ( u ) ES | VSR ly " old eivu dy lim eiyu dy E R - 00

En lyn provided only that the limit in the braces in this last equation

**exists**. Thus ,to complete the proof of the present lemma , it suffices to show that 2 ( y ) 2 ( y ) (

3 ) 0 ( u ) ES | VSR ly " old eivu dy lim eiyu dy E R - 00

**exists**for each u .Page 1733

Then , if f is in H ' ' ( I ) and of is in H ( m ) ( I ) , there

such that the restriction of s to VI belongs to H ( 20 + m ) ( VI ) . This lemma will be

deduced from the following lemma : 20 LEMMA . Let the hypotheses of ...

Then , if f is in H ' ' ( I ) and of is in H ( m ) ( I ) , there

**exists**a neighborhood V of E ,such that the restriction of s to VI belongs to H ( 20 + m ) ( VI ) . This lemma will be

deduced from the following lemma : 20 LEMMA . Let the hypotheses of ...

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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

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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