## Linear Operators: Spectral operators |

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Page 2174

In a Hilbert space the condition that lett ) S M for all te 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 te 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 2312

PROOF . If T * y * = 0 , then y * y = y * Tz = ( T * y * ) 2 = 0 for all y = Tz in the range

of T , and

not in the closure of the range of T , then by the Hahn - Banach theorem ( II . 3 .

PROOF . If T * y * = 0 , then y * y = y * Tz = ( T * y * ) 2 = 0 for all y = Tz in the range

of T , and

**hence**for all y in the closure of the range of T . On the other hand , if y isnot in the closure of the range of T , then by the Hahn - Banach theorem ( II . 3 .

Page 2357

then it is clear that L is a bounded operator and that ( - XI ) - v = ( T – WI ) - " L .

S – AI ) - 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 ( - XI ) - v = ( T – WI ) - " L .

**Hence**( P + N ) ( S – XI ) - V = P ( S – XI ) - " + N ( S – XI ) - = P ( T – 2 ] ) - ' L + N (S – AI ) - V is a bounded operator which is compact if P ( T – 21 ) - " is compact (

cf .

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### Contents

SPECTRAL OPERATORS | 1924 |

Spectral Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

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