## Linear Operators: Spectral operators |

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Page 1954

Q . E . D . It was shown in the course of the preceding proof that for an operator T

with a closed range the point i = 0 is not in the

{ 0 } ' ) X . Thus for all sufficiently small complex numbers 1 # 0 the operator XI ...

Q . E . D . It was shown in the course of the preceding proof that for an operator T

with a closed range the point i = 0 is not in the

**spectrum**of the operator V = T | E ({ 0 } ' ) X . Thus for all sufficiently small complex numbers 1 # 0 the operator XI ...

Page 1955

We shall be concerned with the fine structure of the

points of an operator in X will be classified , as they were in Hilbert space ,

according to the following definition . + 1 DEFINITION . Let A be a bounded linear

...

We shall be concerned with the fine structure of the

**spectrum**, and the spectralpoints of an operator in X will be classified , as they were in Hilbert space ,

according to the following definition . + 1 DEFINITION . Let A be a bounded linear

...

Page 2507

AND 8 uniformly bounded by a constant K in the neighborhood of the

H , while the integrals | | A ( ( A + ie ) I – H , ) - 2012 da and | B * ( ( 1 + ie ) I – H , )

– 1 , 2 da 8 8 8 are bounded as ε →0 . In this case , and under the additional ...

AND 8 uniformly bounded by a constant K in the neighborhood of the

**spectrum**ofH , while the integrals | | A ( ( A + ie ) I – H , ) - 2012 da and | B * ( ( 1 + ie ) I – H , )

– 1 , 2 da 8 8 8 are bounded as ε →0 . In this case , and under the additional ...

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### Contents

SPECTRAL OPERATORS | 1924 |

Spectral Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

31 other sections not shown

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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