## 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 X1 ( 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 ( oo ) X is in D ( $ ( T ) ) if and only if

it is in D ( F ( 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

...

Then by Definition XVIII . 2 . 8 , a vector x in E ( oo ) X is in D ( $ ( T ) ) if and only if

it is in D ( F ( S ) ) , and ,

**moreover**f ( T ) x = 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

...

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### Contents

SPECTRAL OPERATORS 1937 1941 1945 XV Spectral Operators | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

29 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 defined Definition denote dense determined differential operator discrete 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 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