## 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

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 ) , Xy ( 2 ) be the

} , { Ex Ee Bl } , 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 ) = x ...

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

**closed**linear manifolds generated by the sets { EX Ee B} , { Ex Ee Bl } , 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 ) = x ...

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