## Linear Operators, Part 2 |

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Page 1087

Let T , be a 1 - parameter family of bounded operators defined in a subinterval I of

the parameter interval 1 < p 00 , each operator T , acting in the space L ( S , E , u )

.

Let T , be a 1 - parameter family of bounded operators defined in a subinterval I of

the parameter interval 1 < p 00 , each operator T , acting in the space L ( S , E , u )

.

**Suppose**that for P1 , P2 in 1 , T. , and always agree on the intersection of Ln ...Page 1452

PROOF .

21 E ( dx ) arl ? 2 u / * l * 2 e ( T ) , so that if \ ~ \ is bounded , ( Tx , x ) is bounded

below . Conversely ,

PROOF .

**Suppose**that such a y exists . Then , by Theorem XII.2.6 , ( Tx , x ) = S 021 E ( dx ) arl ? 2 u / * l * 2 e ( T ) , so that if \ ~ \ is bounded , ( Tx , x ) is bounded

below . Conversely ,

**suppose**that for each n , en = ( -0 , -n ) no ( T ) is non - void .Page 1597

( 18 ) In the interval [ 0 , 0 ) ,

' ( t ) ) 19 ( t ) ( q ' ( t ) ) 2 ) ( c ) S 1 ° ) dt • đo , for large M. Then the essential

spectrum of r is empty ( Wintner [ 8 ] ) . ( 19 ) In the interval ( a , o ) ,

lim q ...

( 18 ) In the interval [ 0 , 0 ) ,

**suppose**that ( a ) lim q ( t ) -00 , ( b ) lim sup < 00 , ( q' ( t ) ) 19 ( t ) ( q ' ( t ) ) 2 ) ( c ) S 1 ° ) dt • đo , for large M. Then the essential

spectrum of r is empty ( Wintner [ 8 ] ) . ( 19 ) In the interval ( a , o ) ,

**suppose**thatlim q ...

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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