## Linear Operators: Spectral theory |

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Page 993

Then it

. , Qy is independent of V . Q . E . D . 16 THEOREM . If the bounded measurable

function o has its spectral set consisting of the single point m then , for some ...

Then it

**follows**from what has just been demonstrated that Qy , = Qyuy , = Qy , i . e. , Qy is independent of V . Q . E . D . 16 THEOREM . If the bounded measurable

function o has its spectral set consisting of the single point m then , for some ...

Page 996

1 ( d ) it

seen that olf * ) Colp ) and from Lemma 12 ( c ) and the equation of = tf it

that olf * q ) contains no interior point of o ( 9 ) . Hence o ( f * o ) is a closed subset

...

1 ( d ) it

**follows**from the above equation that f * 9 # 0 . From Lemma 12 ( b ) it isseen that olf * ) Colp ) and from Lemma 12 ( c ) and the equation of = tf it

**follows**that olf * q ) contains no interior point of o ( 9 ) . Hence o ( f * o ) is a closed subset

...

Page 1486

If

6 . 35 , it

19 , t has no boundary values . Thus , by Definition 2 . 17 every linear functional ...

If

**follows**immediately that all the coefficients of t are bounded ; thus , by Theorem6 . 35 , it

**follows**that t has no boundary values at + 00 or at – 00 . By Theorem 2 .19 , t has no boundary values . Thus , by Definition 2 . 17 every linear functional ...

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### Contents

BAlgebras | 861 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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additive 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 differential equations domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math matrix measure multiplicity neighborhood norm obtained partial positive preceding present problem Proc projection proof properties prove range regular remark representation respectively restriction result Russian satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero