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

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Results 1-3 of 78

Page 1953

Then since the

Lemma 3 . 1 the operator T is one - to - one . By Theorem II . 2 . 2 , T has a

bounded inverse and hence 0 e p ( T ) = p ( S ) and so SX = X . Next suppose that

E ( { 0 } ...

Then since the

**range**of T is closed , it follows from Corollary 12 that TX = X . ByLemma 3 . 1 the operator T is one - to - one . By Theorem II . 2 . 2 , T has a

bounded inverse and hence 0 e p ( T ) = p ( S ) and so SX = X . Next suppose that

E ( { 0 } ...

Page 1954

Q . E . D . It was shown in the course of the preceding proof that for an operator T

with a closed

{ 0 } ' ) X . Thus for all sufficiently small complex numbers 1 # 0 the operator NI ...

Q . E . D . It was shown in the course of the preceding proof that for an operator T

with a closed

**range**the point i = 0 is not in the spectrum of the operator V = T | E ({ 0 } ' ) X . Thus for all sufficiently small complex numbers 1 # 0 the operator NI ...

Page 2312

Spectral Theory : Self Adjoint Operators in Hilbert Space Nelson Dunford, Jacob

Theodore Schwartz. 7 LEMMA . The closure of the

linear operator T is the set of all x such that y * x = 0 whenever T * y * = 0 . PROOF

.

Spectral Theory : Self Adjoint Operators in Hilbert Space Nelson Dunford, Jacob

Theodore Schwartz. 7 LEMMA . The closure of the

**range**of a densely definedlinear operator T is the set of all x such that y * x = 0 whenever T * y * = 0 . PROOF

.

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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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