## Linear Operators: Spectral theory |

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Page 893

Thus if E is a bounded

operators in the B-space £, and if c e 3, then the integral J f(s) E(ds)w is defined

for every bounded 2-measurable function f on S. It follows immediately that [I.

Thus if E is a bounded

**additive**set function on 2' whose values E(6) are boundedoperators in the B-space £, and if c e 3, then the integral J f(s) E(ds)w is defined

for every bounded 2-measurable function f on S. It follows immediately that [I.

Page 932

Sz.-Nagy's original proof depended on the following theorem due to Neumark [3].

THEOREM. Let S be an abstract set and 2 a field (resp. a-field) of subsets of S.

Let F be an

Sz.-Nagy's original proof depended on the following theorem due to Neumark [3].

THEOREM. Let S be an abstract set and 2 a field (resp. a-field) of subsets of S.

Let F be an

**additive**(resp. weakly countably**additive**) function on 2 to the set of ...Page 958

Hence if e1 and e, are disjoint then p(ei U ex) = E(ei U eg)"p(ei U ex) = [E(e1)+E(

eg)]"p(ei U e2) = E(e)y(ei U ex)+E(eg) p(ei U eg) = p(e1)+(p(eg), so that the

vector valued set function p is

Hence if e1 and e, are disjoint then p(ei U ex) = E(ei U eg)"p(ei U ex) = [E(e1)+E(

eg)]"p(ei U e2) = E(e)y(ei U ex)+E(eg) p(ei U eg) = p(e1)+(p(eg), so that the

vector valued set function p is

**additive**on 30. Therefore, if e1 on e2 = b, the set ...### What people are saying - Write a review

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### Contents

BAlgebras | 859 |

Commutative BAlgebras | 868 |

Commutative BAlgebras | 874 |

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### 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