## Linear Operators: Part III: Spectral Operators [by] Nelson Dunford and Jacob T. Schwartz, with the Assistance of William G. Bade and Robert G. Bartle, Volume 1 |

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Page 2214

5 , the

strongly closed algebra generated by B . Thus every A in W ( B ) is the strong limit

of finite linear combinations of elements of B . It follows that A leaves invariant ...

5 , the

**weakly**closed operator algebra W ( B ) generated by B is the same as thestrongly closed algebra generated by B . Thus every A in W ( B ) is the strong limit

of finite linear combinations of elements of B . It follows that A leaves invariant ...

Page 2217

A bounded linear operator is in the

a o - complete Boolean algebra B of projections in a B - space if and only if it

leaves invariant every closed linear manifold which remains invariant under

every ...

A bounded linear operator is in the

**weakly**closed operator algebra generated bya o - complete Boolean algebra B of projections in a B - space if and only if it

leaves invariant every closed linear manifold which remains invariant under

every ...

Page 2218

as the

uniformly closed operator algebra generated by B1 . Every operator in such a

uniformly closed algebra is , by Lemma 9 , given in terms of a countably additive

spectral ...

as the

**weakly**closed operator algebra generated by B ) is the same as theuniformly closed operator algebra generated by B1 . Every operator in such a

uniformly closed algebra is , by Lemma 9 , given in terms of a countably additive

spectral ...

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### Contents

SPECTRAL OPERATORS XV Spectral Operators | 1924 |

Introduction | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

32 other sections not shown

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### Common terms and phrases

analytic apply arbitrary assumed B-space Banach space Boolean algebra Borel sets boundary conditions bounded bounded Borel bounded operator Chapter clear clearly closure commuting compact complex consider constant contained converges Corollary corresponding countably additive defined Definition denote dense determined differential operator 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 manifold Math Moreover multiplicity norm positive preceding present problem projections PROOF properties prove range regular resolution resolvent respectively restriction Russian satisfies scalar type seen sequence shown shows spectral measure spectral operator spectrum statement strongly subset subspace sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector weakly zero