## Linear Operators, Part 1 |

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Page 364

... the case may be ) which varies continuously with t , and that U = I . 49 Show

that in case An ( x ) = ( 27 ) - 1 / 2qing , the operator Tm is given by the integral

Tmf = lo " ( Uf ) Km ( t ) dt , where Km ( t ) is Km ( 0 , t ) in the notation of

34 .

... the case may be ) which varies continuously with t , and that U = I . 49 Show

that in case An ( x ) = ( 27 ) - 1 / 2qing , the operator Tm is given by the integral

Tmf = lo " ( Uf ) Km ( t ) dt , where Km ( t ) is Km ( 0 , t ) in the notation of

**Exercise**34 .

Page 365

Show that the map : f f defined in

manner onto the closed subspace of L , consisting of those F all of whose

negative Fourier coefficients vanish . 55 Using the notations of

54 , show ...

Show that the map : f f defined in

**Exercise**53 maps H , in a linear one - onemanner onto the closed subspace of L , consisting of those F all of whose

negative Fourier coefficients vanish . 55 Using the notations of

**Exercises**53 and54 , show ...

Page 371

86 Let p , f be as in

let | be a function in H . Then there exists a function g in Ho such that g ( ei ) = 1

for almost all 0 , LEH , , = 1 and such that f / g has no zeros . ( Hint : Generalize ...

86 Let p , f be as in

**Exercise**85 . Then f ( ei ) + 0 for almost all 0 . 87 Let p > 1 andlet | be a function in H . Then there exists a function g in Ho such that g ( ei ) = 1

for almost all 0 , LEH , , = 1 and such that f / g has no zeros . ( Hint : Generalize ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

12 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric neighborhood norm obtained operator positive measure problem Proc PROOF properties proved regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero