## Linear Operators: Spectral theory |

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Page 1027

Hence A

scalar A

have ETa' = Air. Then Ta' = Aw--y, where y

...

Hence A

**belongs**to the spectrum of ET. Conversely, suppose that a non-zeroscalar A

**belongs**to the spectrum of ET. Then, for some non-zero a in EX), wehave ETa' = Air. Then Ta' = Aw--y, where y

**belongs**to the subspace (I–E)S), and...

Page 1116

c, i-1 i=1 so that, by Definition 6.1, B

we let Aq., -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 ...

c, i-1 i=1 so that, by Definition 6.1, B

**belongs**to the Hilbert-Schmidt class Co. Ifwe let Aq., -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 %

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

of the equation (2–1)f = 0 on [0, oo) which is not square-integrable but which ...

Then the point %

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

of the equation (2–1)f = 0 on [0, oo) which is not square-integrable but which ...

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

Copyright | |

48 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 adjoint adjoint operator algebra analytic assume B-algebra basis belongs Borel set boundary conditions boundary values bounded called clear closed closure coefficients commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues element equal equation essential spectrum evident Exercise exists extension fact finite follows formal differential operator formula function function f given Hence Hilbert space ideal identity independent indices inequality integral interval isometric isomorphism Lemma linear mapping matrix measure multiplicity neighborhood norm normal operator obtained positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown singular solution spectral square-integrable statement subset subspace sufficiently Suppose symmetric Theorem theory topology transform unique unit vanishes vector zero