## Linear Operators: Spectral theory |

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Page 1027

... the restriction of the operator ET/(T) to the finite dimensional PRoof. (a) Since

$) is infinite dimensional the origin

Suppose that A # 0

.4.5 ...

... the restriction of the operator ET/(T) to the finite dimensional PRoof. (a) Since

$) is infinite dimensional the origin

**belongs**to the spectrum of both T and ET.Suppose that A # 0

**belongs**to the spectrum of T. Since T is compact, Theorem VII.4.5 ...

Page 1116

co, i-1 i-1 so that, by Definition 6.1, B

we let Ap, = y; "p, then A is plainly self adjoint and A

where r(1–p/2) = p, i.e., r = p(1–p/2)-1. Thus, by Lemma 9, T = BA

class ...

co, i-1 i-1 so that, by Definition 6.1, B

**belongs**to the Hilbert-Schmidt class Cs. Ifwe let Ap, = y; "p, then A is plainly self adjoint and A

**belongs**to the class C.,where r(1–p/2) = p, i.e., r = p(1–p/2)-1. Thus, by Lemma 9, T = BA

**belongs**to theclass ...

Page 1602

Then the point A

]). (48) Suppose that the function q is bounded below, and let f be a real solution

of the equation (A—t)f = 0 on [0, oo) which is not square-integrable but which ...

Then the point A

**belongs**to the essential spectrum of r (Hartman and Wintner [14]). (48) Suppose that the function q is bounded below, and let f be a real solution

of the equation (A—t)f = 0 on [0, oo) which is not square-integrable but which ...

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### Contents

SPECTRAL THEORY | 858 |

Bounded Normal Operators in Hilbert Space | 887 |

Miscellaneous Applications | 937 |

Copyright | |

34 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

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