## Linear operators: Spectral operators |

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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

resolution of the identity defined on the field of all Borel subsets of the plane.

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 2143

This spectral measure is bounded, is

with T. Proof. From Definitions 1, 4, and 7 it is seen that Sf{T) g Sfx(T), and thus for

each 8 in Sf(T) there is, by Lemma 2, one and only one projection E(8) with ...

This spectral measure is bounded, is

**countably additive**on Sf(T), and commuteswith T. Proof. From Definitions 1, 4, and 7 it is seen that Sf{T) g Sfx(T), and thus for

each 8 in Sf(T) there is, by Lemma 2, one and only one projection E(8) with ...

Page 2144

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

show that A(o)A(8)=A{o8). It is seen, by using the above remarks, that for a fixed a

...

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(o)A(8)=A{o8). It is seen, by using the above remarks, that for a fixed a

...

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### Contents

SPECTRAL OPERATORS | 1924 |

Spectral Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

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adjoint operator algebra of projections Amer analytic arbitrary asymptotic B-space Banach space Boolean algebra Borel sets boundary conditions bounded Borel function bounded linear operator bounded operator commuting compact complex numbers complex plane constant contains continuous functions converges Corollary countably additive Definition denote dense differential operator disjoint Doklady Akad domain eigenvalues elements equation exists finite number Foias follows from Lemma follows from Theorem formal differential operator formula Hence Hilbert space hypothesis identity inequality inverse Lebesgue Math matrix measurable functions multiplicity Nauk SSSR norm normal operators operators in Hilbert perturbation polynomial Proof properties prove quasi-nilpotent restriction Russian satisfies scalar operator scalar type operator scalar type spectral Section semi-group sequence shows spectral measure spectral operator spectral theory spectrum strong operator topology subset Suppose trace class type spectral operator unbounded uniformly bounded unique vector weakly complete zero