## Linear Operators: Spectral operators |

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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 Bj . 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 Bj . 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 | 1924 |

Spectral Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

31 other sections not shown

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adjoint operator analytic applications arbitrary assumed B-space Banach space belongs Boolean algebra Borel sets boundary bounded bounded operator Chapter clear closed commuting compact complex condition consider constant contained continuous converges Corollary corresponding countably additive defined Definition denote dense determined domain elements equation equivalent established example exists extension fact finite follows formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator manifold Math Moreover multiplicity norm normal perturbation plane 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 uniformly unique valued vector weakly zero