## Linear Operators: Spectral theory |

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Page 1040

y ( 2 ) is

R ( ā ; T ) * y vanishes which will prove that y ( a ) is

âm , so that y ( a ) can only fail to be

y ( 2 ) is

**analytic**even at a = am . It will now be shown that y2 ( 2 ) = 2N Eām ; T ) *R ( ā ; T ) * y vanishes which will prove that y ( a ) is

**analytic**at all the points a =âm , so that y ( a ) can only fail to be

**analytic**at the point 2 = 0 . To show this ...Page 1102

The determinant det ( I + z7n ) is an

z , if Tn operates in finite - dimensional space , and hence more generally if T ,

has a finite - dimensional range . Thus , since a bounded convergent sequence

of ...

The determinant det ( I + z7n ) is an

**analytic**( and even a polynomial ) function ofz , if Tn operates in finite - dimensional space , and hence more generally if T ,

has a finite - dimensional range . Thus , since a bounded convergent sequence

of ...

Page 1379

If 0 ( • ) is

each Borel set e with compact closure contained in 1 . Thus , by Theorem 25 , 01

, . . . , 0x is a determining set for T . Q . E . D . 28 COROLLARY . Let T , 4 , 0 ; , etc

...

If 0 ( • ) is

**analytic**for j > k , it follows from Theorem 18 that pile ) = 0 for i > k andeach Borel set e with compact closure contained in 1 . Thus , by Theorem 25 , 01

, . . . , 0x is a determining set for T . Q . E . D . 28 COROLLARY . Let T , 4 , 0 ; , etc

...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 869 |

Commutative BAlgebras | 877 |

Copyright | |

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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 remark 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