## 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 2307

The last example above illustrates a problem which will be of great concern to us

in the remainder of this chapter : the problem of finding which formal differential

operators and sets of

The last example above illustrates a problem which will be of great concern to us

in the remainder of this chapter : the problem of finding which formal differential

operators and sets of

**boundary conditions**lead to spectral operators . As our ...Page 2350

But , under the condition that the set of

, this has been established above . Thus , to complete the proof , it suffices to

show that the set of

But , under the condition that the set of

**boundary conditions**Ci ( Uf ) = 0 is regular, this has been established above . Thus , to complete the proof , it suffices to

show that the set of

**boundary conditions**Ci ( Uf ) = 0 , i = 1 , ... , n , is regular .Page 2371

Birkhoff [ 3 ] showed that if the set of

regularity hypotheses of Section 4 , the eigenvalue expansion of a function f of

bounded variation converges to i { f ( t + 0 ) + f ( t - 0 ) } at an interior point t of the

interval ...

Birkhoff [ 3 ] showed that if the set of

**boundary conditions**is subject to theregularity hypotheses of Section 4 , the eigenvalue expansion of a function f of

bounded variation converges to i { f ( t + 0 ) + f ( t - 0 ) } at an interior point t of the

interval ...

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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