## Linear operators: Spectral operators |

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Page 2081

where the supremum is taken over all finite sets {Ej} which are mutually

the sense that Et Ek = 0, j ^ k. Show that there exists a constant K such that v(E,x,x

*)ŁK\x\\x*\. [Hint: Examine the proof of Lemma III. 1.5.] 43 (Berkson) Let E be a ...

where the supremum is taken over all finite sets {Ej} which are mutually

**disjoint**inthe sense that Et Ek = 0, j ^ k. Show that there exists a constant K such that v(E,x,x

*)ŁK\x\\x*\. [Hint: Examine the proof of Lemma III. 1.5.] 43 (Berkson) Let E be a ...

Page 2190

... \at > where the sets alt □ □ □ , cn are

There is an t0 ^ n with E(alo) ^= 0 and l/l* = laJ • If E(aOxo = xo „= 0, then S{f)x0 =

atg x0 and. W)|ŁKI. = I/I«. This establishes the final inequality of the theorem.

... \at > where the sets alt □ □ □ , cn are

**disjoint**sets in E whose union is A.There is an t0 ^ n with E(alo) ^= 0 and l/l* = laJ • If E(aOxo = xo „= 0, then S{f)x0 =

atg x0 and. W)|ŁKI. = I/I«. This establishes the final inequality of the theorem.

Page 2266

A projection E e B will be said to satisfy the countable chain condition if every

family of

denote by # the set of all E e B satisfying this condition. It will be shown that %> is

a ...

A projection E e B will be said to satisfy the countable chain condition if every

family of

**disjoint**projections in B bounded by E is at most countable. We shalldenote by # the set of all E e B satisfying this condition. It will be shown that %> is

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