## Linear Operators: Spectral theory |

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Page 1151

To prove the normality of R we shall use this

disjoint closed sets in R. We select an open set G, in R such that Fin Ki CG1, Gin

F. - %, and then choose an open set H1 such that F., n K. C. H., H., n (Fi U G1) ...

To prove the normality of R we shall use this

**remark**inductively. Let Fı and F, bedisjoint closed sets in R. We select an open set G, in R such that Fin Ki CG1, Gin

F. - %, and then choose an open set H1 such that F., n K. C. H., H., n (Fi U G1) ...

Page 1381

By the

f(1) form a complete set of boundary values for ri and the most general self

adjoint extension To of To(r) is defined by a boundary condition f(0) = e”f(1).

Since [0 ...

By the

**remark**following Definition 2.29, the two linear functionals f –- f(0) and f =>f(1) form a complete set of boundary values for ri and the most general self

adjoint extension To of To(r) is defined by a boundary condition f(0) = e”f(1).

Since [0 ...

Page 1472

We summarize the above

By

defined by the boundary conditions B (if r has boundary values at a) and f(c) ...

We summarize the above

**remarks**for future reference in the following lemma. ...By

**remark**(b) preceding Lemma 41, the adjoint of To of T is the restriction of Ti(r)defined by the boundary conditions B (if r has boundary values at a) and f(c) ...

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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