## Linear Operators: Spectral theory |

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Page 1255

...

f(s)p(s) 81, s2 = (f, g), it follows from the Schwarz inequality that

Consequently, we can define a mapping U(t): 8 -> 98 by letting U(t)a =

...

...

**V**(s+t) for all s and t and (**V**(t)f,**V**(t)g) = X m(s1–s,)f(81–t)g(se—t) =X m(s) +t–s,-t)f(s)p(s) 81, s2 = (f, g), it follows from the Schwarz inequality that

**V**(t)?so C. Qso.Consequently, we can define a mapping U(t): 8 -> 98 by letting U(t)a =

**V**(t)**f-i**-9so,...

Page 1312

Then r(g, —f,) = Tg, —Ti(t)f, = 0, and the functions go-

linearly independent solutions of the equation to = 0. This contradiction

completes the proof. Q.E.D. The next result is a generalization of Corollary 2.26. 2

LEMMA.

Then r(g, —f,) = Tg, —Ti(t)f, = 0, and the functions go-

**fi**, i = 0, 1,...,**v**are**v**-H1linearly independent solutions of the equation to = 0. This contradiction

completes the proof. Q.E.D. The next result is a generalization of Corollary 2.26. 2

LEMMA.

Page 1345

... 9 and Theorem 10, and also of a number of other elementary measure

theoretic properties which these spaces have in common with the spaces L2(u).

However, a few words of caution are in order. If [

.

... 9 and Theorem 10, and also of a number of other elementary measure

theoretic properties which these spaces have in common with the spaces L2(u).

However, a few words of caution are in order. If [

**fi**, ..., f.) ... —co**V**i, j=1 Thus, if [**fi**, f.

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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