## Linear Operators: Spectral operators |

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Page 2194

It has been observed ( cf . VI . 1 . 5 ) that a convex set in the space of all bounded

linear maps between two B - spaces has the same

strong operator topology . Thus the strong and weak operator

It has been observed ( cf . VI . 1 . 5 ) that a convex set in the space of all bounded

linear maps between two B - spaces has the same

**closure**in the weak as in thestrong operator topology . Thus the strong and weak operator

**closures**of an ...Page 2217

Let B be a o - complete Boolean algebra of projections in a B - space X , and let B

, be its strong

Boolean algebra of projections in X . Suppose that B , is not complete .

Let B be a o - complete Boolean algebra of projections in a B - space X , and let B

, be its strong

**closure**. By Lemma 3 , B is bounded and thus B , is also a boundedBoolean algebra of projections in X . Suppose that B , is not complete .

Page 2234

The first statement follows from Definition 8 and the three paragraphs of

explanation which precede it , and from Lemma 6 . Statement ( i ) follows from

Corollary 7 . If e is a bounded Borel set with

supposed ...

The first statement follows from Definition 8 and the three paragraphs of

explanation which precede it , and from Lemma 6 . Statement ( i ) follows from

Corollary 7 . If e is a bounded Borel set with

**closure**contained in U , it may besupposed ...

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### Contents

SPECTRAL OPERATORS | 1924 |

An Operational Calculus for Bounded Spectral | 1941 |

Part | 1950 |

Copyright | |

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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