## 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 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 | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

47 other sections not shown

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adjoint operator Amer analytic apply arbitrary assumed B-space Banach space belongs Boolean algebra Borel set boundary conditions bounded bounded operator Chapter clear closed commuting compact complex constant contains continuous converges Corollary corresponding countably additive defined Definition denote dense determined differential operator domain elements equation equivalent established exists extension fact finite follows formal formula function given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear operator Math Moreover multiplicity norm perturbation plane positive preceding present problem projections PROOF properties prove range resolution resolvent restriction Russian satisfies scalar type seen sequence shown shows similar solution spectral measure spectral operator spectrum subset sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector zero