## Linear Operators, Part 1 |

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Page 358

Show that Snt is given by the formula ( S » ) ( x ) = 1 . ** En ( x , y ) t ( y ) dy , and is

a projection Sn in each of the spaces Ly , BV , CBV , AC , C ( K ) , 1 spoo , k 0 , 1 ,

2 , ... , 00. Show that the range of Sn lies in C ) . 3 Show that Sn +1

Show that Snt is given by the formula ( S » ) ( x ) = 1 . ** En ( x , y ) t ( y ) dy , and is

a projection Sn in each of the spaces Ly , BV , CBV , AC , C ( K ) , 1 spoo , k 0 , 1 ,

2 , ... , 00. Show that the range of Sn lies in C ) . 3 Show that Sn +1

**strongly**in ...Page 685

We recall that the semi - group { T ( t ) , o St } is said to be

dependence upon t is continuous in the strong operator topology , i.e. , if lim 1T ( t

) x — T ( u ) x ] = 0 for each x in X and each tu wadi . u 20. The semi - group is ...

We recall that the semi - group { T ( t ) , o St } is said to be

**strongly**continuous if itsdependence upon t is continuous in the strong operator topology , i.e. , if lim 1T ( t

) x — T ( u ) x ] = 0 for each x in X and each tu wadi . u 20. The semi - group is ...

Page 689

The desired conclusion follows from Corollary II.3.13 . Q.E.D. 3 COROLLARY . In

a reflexive space , the averages A ( Q ) are

bounded and if T ( n ) / n converges to zero

Theorem ...

The desired conclusion follows from Corollary II.3.13 . Q.E.D. 3 COROLLARY . In

a reflexive space , the averages A ( Q ) are

**strongly**convergent if they arebounded and if T ( n ) / n converges to zero

**strongly**. Proof . This follows fromTheorem ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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### Other editions - View all

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

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