## 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 1977

The

by equation ( ii ) is a countably additive spectral measure in HP defined on the

Borel sets B in the complex plane . Since E ( 0 ; Â ( s ) ) and Â ( s ) commute for ...

The

**preceding**theorem shows that the operator valued measure E ( 0 ; A ) givenby equation ( ii ) is a countably additive spectral measure in HP defined on the

Borel sets B in the complex plane . Since E ( 0 ; Â ( s ) ) and Â ( s ) commute for ...

Page 1983

Some Examples of Bounded Spectral Operators In this section some of the

convolution operators in H = L2 ( RN ) , the Hilbert space of square integrable

functions on real ...

Some Examples of Bounded Spectral Operators In this section some of the

**preceding**results will be illustrated by considering the algebra A of allconvolution operators in H = L2 ( RN ) , the Hilbert space of square integrable

functions on real ...

Page 2082

Letting 0 and Y be operators defined on L1 ( 01 , M ) and L . ( 01 , M ) as in the

the non - separable space L . ( 01 , M ) onto a closed separable subspace sp { E (

0 ) ...

Letting 0 and Y be operators defined on L1 ( 01 , M ) and L . ( 01 , M ) as in the

**preceding**exercise , show that Yis a one - to - one and bicontinuous mapping ofthe non - separable space L . ( 01 , M ) onto a closed separable subspace sp { E (

0 ) ...

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