## Linear Operators: Spectral operators |

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Page 2322

1 REGULARITY

and 77 , are of order p . If Regularity

XIX . 4 . 1 XIX . PERTURBATIONS OF SPECTRAL OPERATORS.

1 REGULARITY

**HYPOTHESIS**FOR EVEN ORDER CASE . The polynomials Tryand 77 , are of order p . If Regularity

**Hypothesis**1 is satisfied , we may write 2322XIX . 4 . 1 XIX . PERTURBATIONS OF SPECTRAL OPERATORS.

Page 2342

By the remark following Regularity

loss of generality that Bif = f ... The matrix Ñ ek ( u ) of the remark following

Regularity

0 < k ...

By the remark following Regularity

**Hypothesis**1 , we may then assume withoutloss of generality that Bif = f ... The matrix Ñ ek ( u ) of the remark following

Regularity

**Hypothesis**1 is consequently determined by the equations Ñ * = 0 for0 < k ...

Page 2397

PROOF . The theorem will follow as soon as it is shown that the

Theorem XVIII . ...

7 ( iv ) . It therefore only remains to establish

.

PROOF . The theorem will follow as soon as it is shown that the

**hypotheses**ofTheorem XVIII . ...

**Hypothesis**( ii ) has been established and is given by Lemma7 ( iv ) . It therefore only remains to establish

**hypothesis**( iii ) of Theorem XVIII . 2.

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### Contents

SPECTRAL OPERATORS | 1924 |

Spectral Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

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adjoint operator analytic applications arbitrary assumed B-space Banach space belongs Boolean algebra Borel sets boundary bounded bounded operator Chapter clear closed commuting compact complex condition consider constant contained continuous converges Corollary corresponding countably additive defined Definition denote dense determined domain elements equation equivalent established example exists extension fact finite follows formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator manifold Math Moreover multiplicity norm normal perturbation plane positive preceding present problem projections PROOF properties proved range regular resolution resolvent respectively restriction Russian satisfies scalar type seen sequence shown shows similar spectral measure spectral operator spectrum subset subspace sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector weakly zero