## Linear Operators: Spectral operators |

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Page 2209

The weakly

algebra B which satisfies the condition of the preceding lemma is the same as the

uniformly

the ...

The weakly

**closed**operator algebra generated by a o - complete Booleanalgebra B which satisfies the condition of the preceding lemma is the same as the

uniformly

**closed**algebra generated by B . Proof . It is clear that every element inthe ...

Page 2214

5 , the weakly

strongly

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

Page 2217

Let X ( x ) , X1 ( 2 ) be the

} , { Ex | Ee Bi } , respectively . Since B , is the strong closure of B , each element

Ex with E in B , is contained in X ( x ) and thus X ( x ) 2 X ( 2 ) . Evidently X ( x ) ...

Let X ( x ) , X1 ( 2 ) be the

**closed**linear manifolds generated by the sets { Ex | E B} , { Ex | Ee Bi } , respectively . Since B , is the strong closure of B , each element

Ex with E in B , is contained in X ( x ) and thus X ( x ) 2 X ( 2 ) . Evidently X ( x ) ...

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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