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

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Page 2174

In a Hilbert space the condition that lett | S M for all t e R implies that T is

equivalent to a self adjoint operator and

spectrum . ( This follows from Lemma XV . 6 . 1 which implies that the bounded

group G ...

In a Hilbert space the condition that lett | S M for all t e R implies that T is

equivalent to a self adjoint operator and

**hence**is a scalar type operator with realspectrum . ( This follows from Lemma XV . 6 . 1 which implies that the bounded

group G ...

Page 2295

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

Theodore Schwartz.

X , which shows that o ( T | E ( 0 ; T ' ) X ) Şo . Suppose next that E ( 0 ; T ) = 0 ...

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

Theodore Schwartz.

**Hence**fu ( T ) E ( ; T ' ) X is the inverse of ( uI – T ) | E ( 0 ; T )X , which shows that o ( T | E ( 0 ; T ' ) X ) Şo . Suppose next that E ( 0 ; T ) = 0 ...

Page 2357

then it is clear that L is a bounded operator and that ( 8 – XI ) v = ( T – XI ) - ' L .

S – XI ) - v is a bounded operator which is compact if P ( T - 21 ) - is compact ( cf .

then it is clear that L is a bounded operator and that ( 8 – XI ) v = ( T – XI ) - ' L .

**Hence**( P + N ) ( 8 – XI ) - ° = P ( S – XI ) - ° + N ( S — 2 / ) - = P ( T – XI ) - " L + N (S – XI ) - v is a bounded operator which is compact if P ( T - 21 ) - is compact ( cf .

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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