## 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 2144

It clearly preserves finite disjoint unions , takes complements into complements ,

is countably additive in the X

show that A ( 0 ) A ( S ) = A ( od ) . It is seen , by using the above remarks , that for

...

It clearly preserves finite disjoint unions , takes complements into complements ,

is countably additive in the X

**topology**of X * , and is bounded . It remains only toshow that A ( 0 ) A ( S ) = A ( od ) . It is seen , by using the above remarks , that for

...

Page 2203

The mapping E HO ( E ) is clearly an isomorphism between B and the Boolean

algebra of all open and closed subsets of 1 . These open and closed sets o ( E )

form a basis for the

The mapping E HO ( E ) is clearly an isomorphism between B and the Boolean

algebra of all open and closed subsets of 1 . These open and closed sets o ( E )

form a basis for the

**topology**in 1 . To see this , note that sets of the form { a | | ( T ...Page 2278

If E * € & * we define m ( E * ) , the multiplicity of E * , to be the smallest cardinal

power of a set A of vectors such that E * X * is spanned in the X -

manifolds { N ( x * ) / 2 * € A } . The next lemma establishes the required continuity

...

If E * € & * we define m ( E * ) , the multiplicity of E * , to be the smallest cardinal

power of a set A of vectors such that E * X * is spanned in the X -

**topology**by themanifolds { N ( x * ) / 2 * € A } . The next lemma establishes the required continuity

...

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

28 other sections not shown

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### Common terms and phrases

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 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