## Linear Operators: Spectral theory |

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Page 977

By substituting A ' for 0-9 + ( 5/2 ) and simplifying , it is

-ie ) " lim R- JO S® t ( r ) rer ( * eine ' - ( Ts sin 8 ) ) do ' . 2π Now the Bessel

function Jn of order n is defined by the equation 1 271 J ( 2 ) ei ( n0—2 sin ) d0 ...

By substituting A ' for 0-9 + ( 5/2 ) and simplifying , it is

**seen**that 1 271 G ( u , v ) (-ie ) " lim R- JO S® t ( r ) rer ( * eine ' - ( Ts sin 8 ) ) do ' . 2π Now the Bessel

function Jn of order n is defined by the equation 1 271 J ( 2 ) ei ( n0—2 sin ) d0 ...

Page 1037

product clearly converges to zero for 2 = 2x it is readily

is analytic for a # 0 and vanishes only for a in o ( T ) . It remains to show that if 2 #

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

product clearly converges to zero for 2 = 2x it is readily

**seen**that the function ( T )is analytic for a # 0 and vanishes only for a in o ( T ) . It remains to show that if 2 #

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

Page 1154

Since it is clear that ( 2 ) = Ex £ , what will be proved then , is that ( i ) 2 ( 2 ) ( E ) =

c ( 2x2 ) ( E ) , Εε Σ ( 2 ) , for some constant c independent of E. This condition ( i )

, as is

Since it is clear that ( 2 ) = Ex £ , what will be proved then , is that ( i ) 2 ( 2 ) ( E ) =

c ( 2x2 ) ( E ) , Εε Σ ( 2 ) , for some constant c independent of E. This condition ( i )

, as is

**seen**from Corollary III.11.6 , is a consequence of the assertion that ( ii ) 2 ...### What people are saying - Write a review

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### Contents

BAlgebras | 861 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

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 complete Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense derivatives determined 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 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