## 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 1990

This elementary observation suggests that it might be easier to give certain

convolutions an integral representation if we use the integral of

functions , and this we shall do . We begin by defining the convolution integral (

19 ) ...

This elementary observation suggests that it might be easier to give certain

convolutions an integral representation if we use the integral of

**vector**valuedfunctions , and this we shall do . We begin by defining the convolution integral (

19 ) ...

Page 2160

It will next be shown that the

) X . To see this it will , in view of Corollary II . 3 . 13 , suffice to show that * ( x - y )

= 0 for every linear functional 3 * which vanishes on ( 101 – T ) X . If x * is such a ...

It will next be shown that the

**vector**x - y is in the closure of the manifold ( No I – T) X . To see this it will , in view of Corollary II . 3 . 13 , suffice to show that * ( x - y )

= 0 for every linear functional 3 * which vanishes on ( 101 – T ) X . If x * is such a ...

Page 2266

The cyclic subspace M ( x ) spanned by a

Ee B will be said to satisfy the countable chain condition if every family of disjoint

projections in B bounded by E is at most countable . We shall denote by 6 the set

...

The cyclic subspace M ( x ) spanned by a

**vector**x is sp { Ex Ee B } . A projectionEe B will be said to satisfy the countable chain condition if every family of disjoint

projections in B bounded by E is at most countable . We shall denote by 6 the set

...

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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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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