## Linear Operators: Spectral theory |

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Page 1236

We shall prove the second statement first . As each B ; is a continuous linear

functional on D ( T * ) vanishing on D ( T ) , it is

a closed extension of T. Since the set of boundary conditions is symmetric , it

follows ...

We shall prove the second statement first . As each B ; is a continuous linear

functional on D ( T * ) vanishing on D ( T ) , it is

**clear**from Lemma 5 ( c ) that Ti isa closed extension of T. Since the set of boundary conditions is symmetric , it

follows ...

Page 1243

It is

S ( t ) x --- lim t lim S ( t ) ( e - -Ax . t Thus , if 4 , is the infinitesimal generator of { S (

t ) } , we find that A2 - A . In the same way it is seen that -- AC A. Hence A = -A ...

It is

**clear**that S ( t ) * { S ( t ) } - 1 = U ( t ) for t 2 0. If xe D ( A ) , then ( x - U U ( t ) xS ( t ) x --- lim t lim S ( t ) ( e - -Ax . t Thus , if 4 , is the infinitesimal generator of { S (

t ) } , we find that A2 - A . In the same way it is seen that -- AC A. Hence A = -A ...

Page 1652

Then , since | Flu ) 2 [ F ] , for each F in H ( k ) ( I ) , it is

to some F in Ly ( 1 ) . Similarly , since ( Fl « 2120F \ , for each F in H ( * ) ( I ) and

each index J such that J = k , it is

Then , since | Flu ) 2 [ F ] , for each F in H ( k ) ( I ) , it is

**clear**that { Fn } convergesto some F in Ly ( 1 ) . Similarly , since ( Fl « 2120F \ , for each F in H ( * ) ( I ) and

each index J such that J = k , it is

**clear**that if Jl Sk , the sequence { 2Fn } ...### What people are saying - Write a review

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### Contents

BAlgebras | 859 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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