## Linear Operators: Spectral operators |

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Page 2265

Q . E . D . 3 THEOREM . Let m be a multiplicity function on a complete Boolean

algebra B of projections in a B - space X . Then there is a unique family { En } of

VER .

Q . E . D . 3 THEOREM . Let m be a multiplicity function on a complete Boolean

algebra B of projections in a B - space X . Then there is a unique family { En } of

**disjoint**elements in B , n running over the cardinals < m ( I ) , such that ( i ) I =VER .

Page 2266

A projection Ee B will be said to satisfy the countable chain condition if every

family of

denote by the set of all Ee B satisfying this condition . It will be shown that C is a

dense ...

A projection Ee B will be said to satisfy the countable chain condition if every

family of

**disjoint**projections in B bounded by E is at most countable . We shalldenote by the set of all Ee B satisfying this condition . It will be shown that C is a

dense ...

Page 2267

Finally it must be shown that each EEC is the carrier projection of a vector . Since

EEC we may express E as the union of a sequence of

each of which is the carrier of a vector xn with | xn | 5 1 . Define xo = { n - 1 2 - nxn

.

Finally it must be shown that each EEC is the carrier projection of a vector . Since

EEC we may express E as the union of a sequence of

**disjoint**projections Eneach of which is the carrier of a vector xn with | xn | 5 1 . Define xo = { n - 1 2 - nxn

.

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### Contents

SPECTRAL OPERATORS | 1924 |

An Operational Calculus for Bounded Spectral | 1941 |

Part | 1950 |

Copyright | |

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### Common terms and phrases

adjoint operator analytic apply arbitrary assume B-space Banach space Boolean algebra Borel sets boundary conditions bounded bounded operator Chapter clear closed commuting compact complex consider constant contained continuous converges Corollary corresponding countably additive defined Definition denote dense determined differential operator discrete domain elements equation equivalent established example exists extension fact finite follows formal formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator Math Moreover multiplicity norm positive preceding present problem projections PROOF properties proved range regular resolution resolvent respectively restriction Russian satisfies scalar type seen sequence shown shows similar spectral measure spectral operator spectrum subset subspace sufficiently Suppose Theorem theory topology unbounded uniform uniformly unique valued vector weakly zero