## Linear Operators: Spectral theory |

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Page 889

This means that for every sequence {6,} of disjoint

E ( U 6,)r, a e S). i-1 i-1 A spectral ... on the

iv) for every

This means that for every sequence {6,} of disjoint

**Borel sets**co co (v) X E(6,)a =E ( U 6,)r, a e S). i-1 i-1 A spectral ... on the

**Borel sets**in the plane and satisfying (iv) for every

**Borel set**6 and (v) for every sequence {6,} of disjoint**Borel sets**is ...Page 894

6.2) that a*E(6) r = 0 for every

e 3". It follows (II.3.15) that E(0) = 0. Thus if E and A are bounded additive regular

operator valued set functions defined on the

6.2) that a*E(6) r = 0 for every

**Borel set**Ö in S and every pair w, wo with a e 3, r*e 3". It follows (II.3.15) that E(0) = 0. Thus if E and A are bounded additive regular

operator valued set functions defined on the

**Borel sets**of a normal topological ...Page 913

Let E be the spectral resolution for T and let v, (e) = (E(e)y, , y,) for each

e. ... let {e,} be a sequence of

is a Borel subset of the complement e, of e, and X. v.,(e) = 0, then v.(e) = 0.

Let E be the spectral resolution for T and let v, (e) = (E(e)y, , y,) for each

**Borel set**e. ... let {e,} be a sequence of

**Borel sets**such that X-d v,(e,) = 0, and such that if eis a Borel subset of the complement e, of e, and X. v.,(e) = 0, then v.(e) = 0.

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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