## Linear Operators: Spectral theory |

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Page 868

It is seen from Theorem 1 that for each JJ e -o and each as in 3: there is a

3 into the field p of

this ...

It is seen from Theorem 1 that for each JJ e -o and each as in 3: there is a

**complex number**a (JR) such that a H-S) = w(Jo)e-i-J'. This mapping a -> r(S).) of3 into the field p of

**complex numbers**is clearly a homomorphism. Since w()') s athis ...

Page 902

Then, for every Borel set 6 of

is ... A finite number of these neighborhoods cover the set 6, of those

...

Then, for every Borel set 6 of

**complex numbers**, E,(0) = E(f-(6)). PROOF. Since fis ... A finite number of these neighborhoods cover the set 6, of those

**complex****numbers**ż with |2| < M and whose distance from g(T) is at least 1/n. Thus E(f-'(6,))...

Page 1251

In this section we shall show how the spectral theorem for self adjoint operators

may be applied to prove a number of results from the theory ... For if {m,} has such

a representation, and 20, ..., x, is any finite set of

In this section we shall show how the spectral theorem for self adjoint operators

may be applied to prove a number of results from the theory ... For if {m,} has such

a representation, and 20, ..., x, is any finite set of

**complex numbers**n s m;1,2,3, ...### What people are saying - Write a review

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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