## Linear Operators: Spectral operators |

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Page 2174

In a Hilbert space the condition that lett | M for all te R implies that T is equivalent

to a self adjoint operator and

This follows from Lemma XV.6.1 which implies that the bounded group G ...

In a Hilbert space the condition that lett | M for all te R implies that T is equivalent

to a self adjoint operator and

**hence**is a scalar type operator with real spectrum . (This follows from Lemma XV.6.1 which implies that the bounded group G ...

Page 2312

If T * y * = 0 , then y * y = y * T2 = ( T * y * ) z = 0 for all y = Tz in the range of T ,

and

the closure of the range of T , then by the Hahn - Banach theorem ( II.3.13 ) there

...

If T * y * = 0 , then y * y = y * T2 = ( T * y * ) z = 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 is not inthe closure of the range of T , then by the Hahn - Banach theorem ( II.3.13 ) there

...

Page 2357

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

- 1 ) - " is a bounded operator which is compact if P ( T - AI ) " is compact ( cf. VI.

5.4 ) ...

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

**Hence**( P + N ) ( S – XI ) -v = P ( S – XI ) - " + N ( S – 21 ) - " = P ( T - XI ) -L + N ( S- 1 ) - " is a bounded operator which is compact if P ( T - AI ) " is compact ( cf. VI.

5.4 ) ...

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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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