## 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 1951

Q . E . D 3

E ( o ) with 0 $ 7 . PROOF . By

closed twosided ideal in B ( x ) and so the present

Q . E . D 3

**COROLLARY**. If T is compact , then so are S , N , and every projectionE ( o ) with 0 $ 7 . PROOF . By

**Corollary**V1 . 5 . 5 the compact operators form aclosed twosided ideal in B ( x ) and so the present

**corollary**is immediate .Page 1952

7

by some power of its radical part . PROOF . If T is of finite type , then N " = 0 for

some positive integer n ( Theorem 5 . 4 ) and so TN = 0 . Conversely , if some ...

7

**COROLLARY**. A spectral operator is of finite type if and only if it is annihilatedby some power of its radical part . PROOF . If T is of finite type , then N " = 0 for

some positive integer n ( Theorem 5 . 4 ) and so TN = 0 . Conversely , if some ...

Page 2192

Q . E . D . 12

complete B - space X . Suppose that A is topologically and algebraically

isomorphic to some B - algebra of bounded continuous functions . Then every

operator in A is a ...

Q . E . D . 12

**COROLLARY**. Let A be an algebra of operators in a weaklycomplete B - space X . Suppose that A is topologically and algebraically

isomorphic to some B - algebra of bounded continuous functions . Then every

operator in A is a ...

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### Contents

SPECTRAL OPERATORS XV Spectral Operators | 1924 |

Introduction | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

32 other sections not shown

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