## Linear Operators: Spectral theory |

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Page 1679

Using (1) and (3), we see that to establish the

that (4) g(s,v6)p(-p)f(y)ly)=s, G(v()p(-p))f(y)ly, where G = pH'. Let Ko be a compact

subset of I containing in its interior a second compact set outside of which the ...

Using (1) and (3), we see that to establish the

**present**lemma it suffices to showthat (4) g(s,v6)p(-p)f(y)ly)=s, G(v()p(-p))f(y)ly, where G = pH'. Let Ko be a compact

subset of I containing in its interior a second compact set outside of which the ...

Page 1692

Since we have already seen that lim If,(r)—f,(r) "dr = 0 m, m1-oo JK, for each j, it

follows that lim If,(r)—f,(r) "dr = 0, m, m1-co JI proving the

CoRollary. The conclusions of Corollary 6 and Lemma 8 remain valid even if the

...

Since we have already seen that lim If,(r)—f,(r) "dr = 0 m, m1-oo JK, for each j, it

follows that lim If,(r)—f,(r) "dr = 0, m, m1-co JI proving the

**present**lemma. Q.E.D. 9CoRollary. The conclusions of Corollary 6 and Lemma 8 remain valid even if the

...

Page 1703

In the

partial differential operators to be defined below. A crucial theorem in the

development of the theory of Chapter XIII was Theorem XIII.2.10, which was

based on ...

In the

**present**section it will be seen that it can, at least for the class of ellipticpartial differential operators to be defined below. A crucial theorem in the

development of the theory of Chapter XIII was Theorem XIII.2.10, which was

based on ...

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

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additive Akad algebra Amer analytic applied assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complete Consequently constant contains continuous converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension finite follows formal differential operator formula function function f given Hence Hilbert space identity independent indices inequality integral interval Lemma limit linear mapping Math measure multiplicity neighborhood norm obtained partial positive preceding present problem projection PRoof properties prove range regular remark representation respectively restriction result satisfies seen sequence singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique vanishes vector zero