## Linear Operators: Spectral theory |

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Page 1110

We saw in the course of proving

and by

case in ...

We saw in the course of proving

**Lemma**16 that the function det(I+T) of that**lemma**is continuous in T(cf. the remark which follows**Lemma**16). Thus, by (b)and by

**Lemma**11, to prove (f) in general we have only to prove (f) in the specialcase in ...

Page 1226

Part (a) follows immediately from

from part (a) and

symmetric operator with dense domain has a unique minimal closed symmetric

extension.

Part (a) follows immediately from

**Lemma**5(b), and part (b) follows immediatelyfrom part (a) and

**Lemma**5(c). Q.E.D. It follows from**Lemma**6(b) that anysymmetric operator with dense domain has a unique minimal closed symmetric

extension.

Page 1733

Q.E.D.

neighborhood of the boundary of a domain with smooth boundary. This is carried

out in the next two

differential ...

Q.E.D.

**Lemma**18 enables us to use the method of proof of Theorem 2 in theneighborhood of the boundary of a domain with smooth boundary. This is carried

out in the next two

**lemmas**. 19**LEMMA**. Leto be an elliptic formal partialdifferential ...

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### Contents

SPECTRAL THEORY | 858 |

868 | 885 |

Miscellaneous Applications | 937 |

Copyright | |

38 other sections not shown

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additive Akad algebra Amer analytic assume Banach spaces basis belongs Borel boundary conditions boundary values bounded called clear closed closure coefficients compact complex 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 matrix 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