## Linear Operators, Part 2 |

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Page 1083

Consequently , the series 4 ( 2 ) — 20 ( 2 ) 1 = 24 , 2 " N = 2 of the preceding

measure space . Then an operator A in the Hilbert space L ( S , & , u ) is of Hilbert

...

Consequently , the series 4 ( 2 ) — 20 ( 2 ) 1 = 24 , 2 " N = 2 of the preceding

**exercise**converges in the Hilbert - Schmidt norm . 44 Let ( S , E , u ) be a positivemeasure space . Then an operator A in the Hilbert space L ( S , & , u ) is of Hilbert

...

Page 1086

Show , finally , that by choosing A ( s , 3 ) = 0 for all s in S , we obtain the result of

method of

of ...

Show , finally , that by choosing A ( s , 3 ) = 0 for all s in S , we obtain the result of

**Exercise**46 as a special case of the present result . ( Hint : Generalize themethod of

**Exercise**46 . ) 49 The operator A of Hilbert - Schmidt class is said to beof ...

Page 1087

( Hint : For ( d ) , use Weyl ' s inequality ,

a l - parameter family of bounded operators defined in a subinterval 1 of the

parameter ...

( Hint : For ( d ) , use Weyl ' s inequality ,

**Exercise**30 . ) E . Miscellaneous**Exercises**50 ( Halberg ) Let ( S , E , u ) be a o - finite measure space . Let T . bea l - parameter family of bounded operators defined in a subinterval 1 of the

parameter ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

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additive adjoint operator 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 derivatives determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function give 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 satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero