## Linear Operators: Spectral theory |

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Page 1092

Let & be a finite-dimensional space including both the

of To; suppose that the dimension of & is d. Then, plainly, & is invariant under

Tand T*, and, since (Té", w)=(&", Toa.) = 0 for all a, we have T&H = 0 and similarly

...

Let & be a finite-dimensional space including both the

**range**of T and the**range**of To; suppose that the dimension of & is d. Then, plainly, & is invariant under

Tand T*, and, since (Té", w)=(&", Toa.) = 0 for all a, we have T&H = 0 and similarly

...

Page 1395

Then (E(01)U)a' = (I–E({2})(AI–T))a' = (AI-T), which shows that the

projection E(01) contains the

disjoint from ol, and let f(u) = (A-u)" if u # V and f(u) = 0 if u e V. Suppose that y is

in ...

Then (E(01)U)a' = (I–E({2})(AI–T))a' = (AI-T), which shows that the

**range**of theprojection E(01) contains the

**range**of T. Choose a neighborhood V of A which isdisjoint from ol, and let f(u) = (A-u)" if u # V and f(u) = 0 if u e V. Suppose that y is

in ...

Page 1397

(T) implies that the

easily seen to be symmetric — obtained by restricting To to 3 (T)+9°. Then the

the ...

(T) implies that the

**range**R(T) of T is closed. Let T1 be the extension — which iseasily seen to be symmetric — obtained by restricting To to 3 (T)+9°. Then the

**range**R(T) of Ti coincides with the**range**of T and is therefore closed. Moreover,the ...

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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