## Linear Operators: Spectral operators |

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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 1931

+ 4 COROLLARY . If the domain of a

- field , then E is

The boundedness of E ( o ) follows from Corollaries IV . 10 . 2 and II . 3 . 21 .

+ 4 COROLLARY . 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

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

...

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### Contents

SPECTRAL OPERATORS | 1924 |

An Operational Calculus for Bounded Spectral | 1941 |

Part | 1950 |

Copyright | |

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### Common terms and phrases

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