## Linear Operators: Part III: Spectral Operators [by] Nelson Dunford and Jacob T. Schwartz, with the Assistance of William G. Bade and Robert G. Bartle, Volume 1 |

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Page 1930

However , in Corollary X.2.4 we have seen that a bounded normal operator in Hilbert space always has a uniquely defined bounded and

However , in Corollary X.2.4 we have seen that a bounded normal operator in Hilbert space always has a uniquely defined bounded and

**countably additive**resolution of the identity defined on the field of all Borel subsets of the plane .Page 1931

If the domain of a

If the domain of a

**countably additive**spectral measure E is a o - field , then E is**countably additive**in the strong operator topology and bounded . The boundedness of E ( o ) follows from Corollaries IV.10.2 and II.3.21 .Page 2144

It clearly preserves finite disjoint unions , takes complements into complements , is

It clearly preserves finite disjoint unions , takes complements into complements , is

**countably additive**in the X topology of X * , and is bounded . It remains only to show that A ( 0 ) A ( S ) = A ( 08 ) .### What people are saying - Write a review

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

47 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 defined Definition denote dense determined differential operator discrete 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