## Linear Operators: Spectral operators |

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Page 2188

Let E be a spectral measure in the complex B - space X which is defined and

countably additive on a o - field of subsets of a set 1 and let g be a bounded

measurable function defined on the complex plane . Then $ 9 ( f ( n ) ) E ( da ) = g

...

Let E be a spectral measure in the complex B - space X which is defined and

countably additive on a o - field of subsets of a set 1 and let g be a bounded

**Borel**measurable function defined on the complex plane . Then $ 9 ( f ( n ) ) E ( da ) = g

...

Page 2233

The operator f ( T ) of Definition 8 is closed , linear , and independent of the

particular sequence of

each x in D ( f ( T ) ) , E ( e ) D ( $ ( T ) ) s D ( $ ( T ) ) and E ( e ) f ( T ) x = f ( T ) E (

e ) x .

The operator f ( T ) of Definition 8 is closed , linear , and independent of the

particular sequence of

**Borel**sets used to define it . ( i ) For each**Borel**set e andeach x in D ( f ( T ) ) , E ( e ) D ( $ ( T ) ) s D ( $ ( T ) ) and E ( e ) f ( T ) x = f ( T ) E (

e ) x .

Page 2262

Furthermore , if equation ( 5 ) holds for each function in a uniformly bounded

pointwise convergent sequence { fn } , then it follows from ( 4 ) that it holds for the

limit function f = lim fr . no Hence ( 5 ) holds if f is any bounded

Furthermore , if equation ( 5 ) holds for each function in a uniformly bounded

pointwise convergent sequence { fn } , then it follows from ( 4 ) that it holds for the

limit function f = lim fr . no Hence ( 5 ) holds if f is any bounded

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### Contents

SPECTRAL OPERATORS | 1924 |

An Operational Calculus for Bounded Spectral | 1941 |

Part | 1950 |

Copyright | |

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