## Linear operators: Spectral operators |

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Page 1947

Then there exists a bounded self

everywhere defined inverse such that BEB'1 is a self adjoint projection for every

E in the Boolean algebra determined by the algebras JBj, . . . , SBk . Proof. For E

e 33( ...

Then there exists a bounded self

**adjoint operator**B in § with a boundedeverywhere defined inverse such that BEB'1 is a self adjoint projection for every

E in the Boolean algebra determined by the algebras JBj, . . . , SBk . Proof. For E

e 33( ...

Page 2169

Then r({T){i;I — T) =E(a), which shows that (T | E(a)X) £ a and proves that T is a

spectral

the ...

Then r({T){i;I — T) =E(a), which shows that (T | E(a)X) £ a and proves that T is a

spectral

**operator**. It follows from (vii) that T is a scalar**operator**. Q.E.D. 6. Self**Adjoint**Operators in Hilbert Space It is the purpose of this section to show howthe ...

Page 2170

These lemmas will show that the hypotheses of Theorem 5.18 are satisfied by a

self

These lemmas will show that the hypotheses of Theorem 5.18 are satisfied by a

self

**adjoint operator**in Hilbert space. 1 Lemma. If T is a bounded self**adjoint****operator**in Hilbert space, its spectrum is real and for every non-real a. we have ...### What people are saying - Write a review

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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