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

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Page 1940

8 that S is uniquely

T . It follows from Lemma 4 that T and S have the same spectrum . Next it will be

shown that every spectral operator T has the desired decomposition .

8 that S is uniquely

**determined**by T . Hence N = T - S is uniquely**determined**byT . It follows from Lemma 4 that T and S have the same spectrum . Next it will be

shown that every spectral operator T has the desired decomposition .

Page 2029

10 DEFINITION . A scalar function y on RN is said to

distribution T , in RN if for every p in 0 , ( s ) ( s ) is Lebesgue integrable on RN

and the functional Tv defined by the equation ( 49 ) Tv ( q ) = 0 ( 8 ) ( s ) ds , deb ,

is ...

10 DEFINITION . A scalar function y on RN is said to

**determine**the tempereddistribution T , in RN if for every p in 0 , ( s ) ( s ) is Lebesgue integrable on RN

and the functional Tv defined by the equation ( 49 ) Tv ( q ) = 0 ( 8 ) ( s ) ds , deb ,

is ...

Page 2373

which exclude the whole complex plane from the spectrum of the operator they

operators having eigenvalues d = 82

ks .

which exclude the whole complex plane from the spectrum of the operator they

**determine**. ... These boundary conditions**determine**discrete differentialoperators having eigenvalues d = 82

**determined**by equations of the form sin 8 =ks .

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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