## Linear Operators: Spectral operators |

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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

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 .Page 2233

9 THEOREM . The operator f ( T ) of Definition 8 is closed , linear , and independent of the particular sequence of

9 THEOREM . 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**sete and each x in D ( f ( T ) ) , E ( e ) D ( $ ( T ) ) S DIF ( T ) ...Page 2262

Furthermore , if equation ( 5 ) holds for each function in a uniformly bounded pointwise convergent sequence { n } , then it follows from ( 4 ) that it holds for the limit function f = limfn Hence ( 5 ) holds if f is any bounded

Furthermore , if equation ( 5 ) holds for each function in a uniformly bounded pointwise convergent sequence { n } , then it follows from ( 4 ) that it holds for the limit function f = limfn Hence ( 5 ) holds if f is any bounded

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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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adjoint operator Amer analytic apply arbitrary assumed B-space Banach space belongs Boolean algebra Borel set boundary conditions bounded bounded operator Chapter clear closed commuting compact complex constant contains continuous converges Corollary corresponding countably additive defined Definition denote dense determined differential operator domain elements equation equivalent established exists extension fact finite follows formal formula function given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear operator Math Moreover multiplicity norm perturbation plane positive preceding present problem projections PROOF properties prove range resolution resolvent restriction Russian satisfies scalar type seen sequence shown shows similar solution spectral measure spectral operator spectrum subset sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector zero