## Linear Operators: Self Adjoint Operators in Hilbert Space. Spectral theory. Part II |

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Page 1037

product clearly converges to zero for 1 = 2x it is readily seen that the function Pi (

T ) is

a + 0 , then P. ( T ) is continuous in T relative to the Hilbert - Schmidt norm in HS .

product clearly converges to zero for 1 = 2x it is readily seen that the function Pi (

T ) is

**analytic**for a # 0 and vanishes only for 2 in o ( T ) . It remains to show that ifa + 0 , then P. ( T ) is continuous in T relative to the Hilbert - Schmidt norm in HS .

Page 1040

yı ( 2 ) is

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

so that y ( 2 ) can only fail to be

yı ( 2 ) is

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

**analytic**at all the points à = Am ,so that y ( 2 ) can only fail to be

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

The determinant det ( I + zTn ) is an

2 , 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 + zTn ) is an

**analytic**( and even a polynomial ) function of2 , 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 ...

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### Contents

SPECTRAL THEORY Self Adjoint Operators in Hilbert Space | 858 |

BAlgebras | 859 |

Preliminary Notions | 865 |

Copyright | |

61 other sections not shown

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### Common terms and phrases

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