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

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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 1956

Q . E . D . 3 THEOREM . If T is of finite type , its residual

point , is in its point

spectral point of T . If E ( { 2 } ) # 0 then E ( { 2 } ) x = x for some x = 0 and No = 0

for some ...

Q . E . D . 3 THEOREM . If T is of finite type , its residual

**spectrum**is void and apoint , is in its point

**spectrum**if and only if E ( { 2 } ) + 0 . PROOF . Let , be aspectral point of T . If E ( { 2 } ) # 0 then E ( { 2 } ) x = x for some x = 0 and No = 0

for some ...

Page 1957

Thus , by the preceding corollary , we have O ( S ) Şo ( To ) , and so to prove the

present corollary , it suffices to prove that is in the continuous

( S – 11 ) x = 0 , where x is in E ( 0 ) X . Since S and T have the same resolution ...

Thus , by the preceding corollary , we have O ( S ) Şo ( To ) , and so to prove the

present corollary , it suffices to prove that is in the continuous

**spectrum**of S . . Let( S – 11 ) x = 0 , where x is in E ( 0 ) X . Since S and T have the same resolution ...

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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 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