## Linear Operators: Part III: Spectral Operators [by] Nelson Dunford and Jacob T. Schwartz, with the Assistance of William G. Bade and Robert G. Bartle, Volume 1 |

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Page 2064

4 that ( 5 ) lal slal + 1f | 1 , where Ifl , is the Lį

complete as a subalgebra of A , but if we renorm it by placing lalı = lal + 1f | 1 ,

then the completeness of Lį assures the completeness of U , under the

and ...

4 that ( 5 ) lal slal + 1f | 1 , where Ifl , is the Lį

**norm**of f . The algebra A4 is notcomplete as a subalgebra of A , but if we renorm it by placing lalı = lal + 1f | 1 ,

then the completeness of Lį assures the completeness of U , under the

**norm**( 6 )and ...

Page 2070

Theorem 1 shows that the operator a determines a and f uniquely , and so we

may define a

that A . is a B - space under this

...

Theorem 1 shows that the operator a determines a and f uniquely , and so we

may define a

**norm**in A , by the equation ( 18 ) lalo = lal + iflo , ae A . . It is clearthat A . is a B - space under this

**norm**. It is also an algebra , for the product of two...

Page 2462

Moreover , if C belongs to the trace class C1 , then T , C converges to zero in

trace

xe H | 1 } ) is conditionally compact , and thus for each € > 0 there exists a finite

set ...

Moreover , if C belongs to the trace class C1 , then T , C converges to zero in

trace

**norm**, and CT * converges to zero in trace**norm**. PROOF . The set K = C ( {xe H | 1 } ) is conditionally compact , and thus for each € > 0 there exists a finite

set ...

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

28 other sections not shown

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