## 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 1917

If an unperturbed operator T and a perturbed operator T ” derived from it are both

self adjoint , then a non - rigorous

expectation that the Friedrichs operator U realizing the similarity UTU - 1 = T ...

If an unperturbed operator T and a perturbed operator T ” derived from it are both

self adjoint , then a non - rigorous

**formal**argument can be used to buttress theexpectation that the Friedrichs operator U realizing the similarity UTU - 1 = T ...

Page 2020

We may therefore conclude from Theorem 7 that the natural closed extension As

of the

whole complex plane unless A is of order zero , that is , none of its elements ...

We may therefore conclude from Theorem 7 that the natural closed extension As

of the

**formal**differential operator ( 18 ) with An P = 0 has its spectrum o ( As ) thewhole complex plane unless A is of order zero , that is , none of its elements ...

Page 2318

Q . E . D . 12 COROLLARY . Let q be a bounded measurable function and T be

the unbounded differential operator defined by a

12 + 9 ( 2 ) dc and by the boundary conditions [ * ] . Then T is a spectral operator .

Q . E . D . 12 COROLLARY . Let q be a bounded measurable function and T be

the unbounded differential operator defined by a

**formal**differential operator 10 .12 + 9 ( 2 ) dc and by the boundary conditions [ * ] . Then T is a spectral operator .

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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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### 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 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