## Linear Operators: Spectral operators |

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Page 2239

If x is in E ( ē ) X as well as in E ( e ) X , it

If x is in E ( ē ) X as well as in E ( e ) X , it

**follows**from the operational calculus for bounded functions ( cf. ... well defined on Veeso Ele ) X . It thus**follows**from Lemma 6 that T ( f ) is a closed , densely defined operator .Page 2246

Let Pin it

Let Pin it

**follows**that R ( a ) is a bounded operator whose range is contained in the domain of C. It is clear then that ( ^ I – C ) R ( ) . 2 for x in H and R ( A ) ( A -C ) x = x for x in D ( C ) , so that R ( A ) = R ( A ; C ) and 1 ...Page 2459

Statement ( b ) of our lemma

Statement ( b ) of our lemma

**follows**at once . ... Thus statement ( a ) of our theorem**follows**. ... If ( il – H ) - Lac ( H ) were not dense in Lac ( H ) , it would**follow**by the Hahn - Banach theorem ( cf. Corollary II.3.13 ) that for ...### What people are saying - Write a review

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### Contents

SPECTRAL OPERATORS 1937 1941 1945 XV Spectral Operators | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

32 other sections not shown

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adjoint operator Amer analytic apply arbitrary assumed B-space Banach space belongs Boolean algebra Borel set boundary conditions bounded bounded operator Chapter clear closed commuting compact complex constant contains continuous converges Corollary corresponding countably additive defined Definition denote dense determined differential operator domain elements equation equivalent established exists extension fact finite follows formal formula function given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear operator Math Moreover multiplicity norm perturbation plane positive preceding present problem projections PROOF properties prove range resolution resolvent restriction Russian satisfies scalar type seen sequence shown shows similar solution spectral measure spectral operator spectrum subset sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector zero