## Linear Operators: General theory |

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Page 304

T : L → L , is

Similarly T 2T , or T , STį means that T1 - T , 2 0 . It is readily seen that if T 20 ,

then T maps real functions into real functions . 25 LEMMA . Let ( S , E , u ) be a

T : L → L , is

**positive**and write T 2 0 when f e L , and í 2 0 imply that Tj 20 .Similarly T 2T , or T , STį means that T1 - T , 2 0 . It is readily seen that if T 20 ,

then T maps real functions into real functions . 25 LEMMA . Let ( S , E , u ) be a

**positive**...Page 305

We now show that T , is additive on

20 , i = 1 , 2 , and let f1 = 21 – 1811 , tz = - 1 & 2i be decompositions of fı and f ,

into

+ ...

We now show that T , is additive on

**positive**elements . Let to € L ( S , E , u ) , to20 , i = 1 , 2 , and let f1 = 21 – 1811 , tz = - 1 & 2i be decompositions of fı and f ,

into

**positive**functions . Then dit & 2i is a decomposition of f1 + 12 , and so To ( 41+ ...

Page 714

( a ) there exists a

exponentially fast , for x € Co . Moreover , the subspaces Cp and C , are uniquely

defined by properties ( a ) and ( b ) , respectively . Proof . It follows from its

definition ...

( a ) there exists a

**positive**integer N such that TNx = x , for xe Cp ; ( b ) Tnx = 0exponentially fast , for x € Co . Moreover , the subspaces Cp and C , are uniquely

defined by properties ( a ) and ( b ) , respectively . Proof . It follows from its

definition ...

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### Contents

Metric Spaces | 19 |

Convergence and Uniform Convergence of Generalized | 26 |

Exercises | 33 |

Copyright | |

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