## Linear Operators: Spectral theory |

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Page 1463

is an evident consequence of

the pair of functions sin At and sin ut for u > A; the corollary by the pair of

functions ...

**Statement**(a) is the special case ti = 12 of the preceding theorem.**Statement**(b)is an evident consequence of

**statement**(a). Q.E.D. The theorem is illustrated bythe pair of functions sin At and sin ut for u > A; the corollary by the pair of

functions ...

Page 1469

The operator r is bounded below if and only if it is finite below A for some finite A.

Thus

) ...

The operator r is bounded below if and only if it is finite below A for some finite A.

Thus

**statement**(a) follows immediately from the preceding lemma. To prove**statement**(b), note that t is finite below Żo, but not below any A - Wo.**Statement**(b) ...

Page 1771

... a e I, such that u(t, x) = 6,(2)u(t, r) = ... = 8*T*(X)u(t, w) = 0, t > 0, a e2, and such

that in us.')-10-0. PRoof.

Theorem 6.23.

...

... a e I, such that u(t, x) = 6,(2)u(t, r) = ... = 8*T*(X)u(t, w) = 0, t > 0, a e2, and such

that in us.')-10-0. PRoof.

**Statement**(i) follows from the preceding theorem andTheorem 6.23.

**Statement**(ii) follows from**statement**(ii) of the preceding theorem,...

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### Contents

SPECTRAL THEORY | 858 |

868 | 885 |

Miscellaneous Applications | 937 |

Copyright | |

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