## Linear Operators: Spectral operators |

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Page 2381

It remains to verify the

which we have constructed and for its derivative a[(t, it). Given e > 0, we note,

using (7), that (11) o-(<,tt)e-«"-l=A„(0-l = (£A)(0, Cf*)e[«. °o)xP,+ . Moreover, by (3)

, (12) ...

It remains to verify the

**asymptotic**properties asserted for the solution o'iC. y)which we have constructed and for its derivative a[(t, it). Given e > 0, we note,

using (7), that (11) o-(<,tt)e-«"-l=A„(0-l = (£A)(0, Cf*)e[«. °o)xP,+ . Moreover, by (3)

, (12) ...

Page 2394

It is clear from Lemma 1 that such a linear combination can only have the

indicated

(A))a1(«, ,*(A)). Therefore B + (\)=A~(X) + c{(J.(X))A + (X), and A + (X)o2(s, fi{X)) ...

It is clear from Lemma 1 that such a linear combination can only have the

indicated

**asymptotic**form if o = l,6= c(ix(A)). Hence °S, /*(A)) = <J1(t, -«*(A)) + c(/x(A))a1(«, ,*(A)). Therefore B + (\)=A~(X) + c{(J.(X))A + (X), and A + (X)o2(s, fi{X)) ...

Page 2399

Then the equation to = 0 has two solutions a1 and a2 satisfying the

relationships aj(<) ~ 1, a2(t) ~ t as t-> oo. Proof. We saw in Corollary 2 that a^t) =

a^t, 0) satisfies the first of these

t) ...

Then the equation to = 0 has two solutions a1 and a2 satisfying the

**asymptotic**relationships aj(<) ~ 1, a2(t) ~ t as t-> oo. Proof. We saw in Corollary 2 that a^t) =

a^t, 0) satisfies the first of these

**asymptotic**relationships. Let a be so large that a^t) ...

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### Contents

SPECTRAL OPERATORS | 1924 |

Spectral Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

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