## Linear Operators: Spectral operators |

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Page 2094

Berkson and Dowson [ 1 ] have considered prespectral operators in some detail

and have obtained a number of results concerning them . They show that if T is

spectral , then 7 * has a unique resolution of the identity of class ( X ) ;

if ...

Berkson and Dowson [ 1 ] have considered prespectral operators in some detail

and have obtained a number of results concerning them . They show that if T is

spectral , then 7 * has a unique resolution of the identity of class ( X ) ;

**moreover**,if ...

Page 2115

F if and only if T and U are quasi - nilpotent equivalent . If T is a spectral operator

and T and U are quasi - nilpotent equivalent , then U is a spectral operator .

**Moreover**, if T and U are decomposable , then X7 ( F ) = xy ( F ) for all closed setsF if and only if T and U are quasi - nilpotent equivalent . If T is a spectral operator

and T and U are quasi - nilpotent equivalent , then U is a spectral operator .

Page 2356

Then by Definition XVIII . 2 . 8 , a vector x in E ( có ) X is in Dif ( T ) ) if and only if it

is in DIF ( S ) ) , and ,

defined on o ( T ) and for x e E ( 0 . ) X . In particular , x is in D ( T ) if and only if x

is in D ...

Then by Definition XVIII . 2 . 8 , a vector x in E ( có ) X is in Dif ( T ) ) if and only if it

is in DIF ( S ) ) , and ,

**moreover**f ( T ) = f ( S ) x , for each analytic function fdefined on o ( T ) and for x e E ( 0 . ) X . In particular , x is in D ( T ) if and only if x

is in D ...

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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