## Linear Operators: Spectral Theory : Self Adjoint Operators in Hilbert Space, Volume 2 |

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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

6

= 0 if 0 € 7 ( respectively SA . = NA . = E ( 0 ) A , = 0 if 0 € 7 ) . 7

spectral operator is of finite type if and only if it is annihilated by some power of ...

6

**COROLLARY**. If A , T = 0 ( respectively TA , = 0 ) , then A , S = A , N = A , E ( o )= 0 if 0 € 7 ( respectively SA . = NA . = E ( 0 ) A , = 0 if 0 € 7 ) . 7

**COROLLARY**. Aspectral operator is of finite type if and only if it is annihilated by some power of ...

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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### Common terms and phrases

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 formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator manifold Math Moreover multiplicity norm normal 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