## Linear operators: Spectral operators |

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Page 1994

Let Xn<p

Theorem 4, Xn9>= F~l\x> an^ since Xn<p

of F that lim f |An(S)-Xm(*)|2(fo = 0, m,n— *co o which implies (III. 3. 6) that \n

Let Xn<p

**converge**for each <p in §. Theorem 4 shows that |X„I ... Then, byTheorem 4, Xn9>= F~l\x> an^ since Xn<p

**converges**, we see from the continuityof F that lim f |An(S)-Xm(*)|2(fo = 0, m,n— *co o which implies (III. 3. 6) that \n

**converges**...Page 2218

It must be shown that {Ea}

a consideration of the sequence {Ea — E) shows that it may be assumed that E =

0. Thus, to make an indirect proof, it is assumed that the sequence {Ea} is ...

It must be shown that {Ea}

**converges**strongly to E. By Lemma 6, E is in B and soa consideration of the sequence {Ea — E) shows that it may be assumed that E =

0. Thus, to make an indirect proof, it is assumed that the sequence {Ea} is ...

Page 2462

Moreover, ifC belongs to the trace class <€x, then TnC

norm, and CT*

1 }) is conditionally compact, and thus for each e > 0 there exists a finite set xlt ...

Moreover, ifC belongs to the trace class <€x, then TnC

**converges**to zero in tracenorm, and CT*

**converges**to zero in trace norm. Proof. The set K = C({x e § [ |x| ^1 }) is conditionally compact, and thus for each e > 0 there exists a finite set xlt ...

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### Contents

SPECTRAL OPERATORS | 1924 |

Spectral Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

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