## Linear Operators: Spectral Theory : Self Adjoint Operators in Hilbert Space, Volume 2 |

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Page 2209

The weakly

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

uniformly

in ...

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 elementin ...

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 ( x ) be the

} , { Ex | E 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 X7 ( x ) . Evidently X ( x ) X ...

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

**closed**linear manifolds generated by the sets { Ex Ee B} , { Ex | E 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 X7 ( x ) . Evidently X ( x ) X ...

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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 formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator manifold Math Moreover multiplicity norm normal 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