## 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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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

The closure of the

such that y * x = 0 whenever T * y * = 0 . ...

closure of the

...

The closure of the

**range**of a densely defined linear operator T is the set of all xsuch that y * x = 0 whenever T * y * = 0 . ...

**range**of T , and hence for all y in theclosure of the

**range**of T . On the other hand , if y is not in the closure of the**range**...

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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