## Linear Operators: Spectral theory |

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Page 1040

yı ( ) is

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

that y ( 2 ) can only fail to be

yı ( ) is

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

**analytic**at all the points a = hm , sothat y ( 2 ) can only fail to be

**analytic**at the point i = 0. To show this , note that ...Page 1102

The determinant det ( I + zT , ) is an

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

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

of ...

The determinant det ( I + zT , ) is an

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

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

of ...

Page 1492

... the matrix B ( A ) has a fixed number k of eigenvalues which are given by

functions Pi ( 2 ) , ... , 8x ( 2 )

in U in which two or more of these distinct

value . ) ...

... the matrix B ( A ) has a fixed number k of eigenvalues which are given by

functions Pi ( 2 ) , ... , 8x ( 2 )

**analytic**in U. ( The set e , is that isolated set of pointsin U in which two or more of these distinct

**analytic**functions take on the samevalue . ) ...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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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 neighborhood norm obtained partial positive preceding present problem 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