## 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

4 we have seen that a bounded normal operator in Hilbert space always has a

uniquely defined bounded and

defined on the field of all Borel subsets of the plane . The operators we shall

study in the ...

4 we have seen that a bounded normal operator in Hilbert space always has a

uniquely defined bounded and

**countably additive**resolution of the identitydefined on the field of all Borel subsets of the plane . The operators we shall

study in the ...

Page 2144

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

is

show that A ( 0 ) A ( S ) = A ( od ) . It is seen , by using the above remarks , that for

...

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 toshow that A ( 0 ) A ( S ) = A ( od ) . It is seen , by using the above remarks , that for

...

Page 2591

(1929) integral with respect to, XV.2 (1929) Spectral operator, adjoint of, XVI.4.6 (

2148), XVI.5. 16 (2162) canonical reduction of, XV.4 (1937), XV.4.5 (1939), ...

**countably additive**, XV.2 (1930), XV.2.3 (1930), XV.2.4 (1931) definition of, XV.2.1(1929) integral with respect to, XV.2 (1929) Spectral operator, adjoint of, XVI.4.6 (

2148), XVI.5. 16 (2162) canonical reduction of, XV.4 (1937), XV.4.5 (1939), ...

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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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### Common terms and phrases

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