## Linear Operators: Spectral operators |

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Page 2169

This shows that (vi) holds for every

continuous function g. A repetition of this argument shows that it also holds if/ and

g are both

and also, ...

This shows that (vi) holds for every

**bounded**Borel function / and everycontinuous function g. A repetition of this argument shows that it also holds if/ and

g are both

**bounded**Borel functions. Thus the**operators**f(T) and g(T) commuteand also, ...

Page 2239

Since/x,, is a bounded function, the operator T(fXe) is a

in E(e)X as well as in E(e)X, it follows from the operational calculus for bounded

functions (cf. XVII.2.10) that T(fXe)x = T(fXe)E(e)x = T(fXel)x = T(fXi)E(e)x = T(fXl)x

...

Since/x,, is a bounded function, the operator T(fXe) is a

**bounded operator**. If a; isin E(e)X as well as in E(e)X, it follows from the operational calculus for bounded

functions (cf. XVII.2.10) that T(fXe)x = T(fXe)E(e)x = T(fXel)x = T(fXi)E(e)x = T(fXl)x

...

Page 2252

An attempt to follow the development in the

runs into difficulties. The

quasi-nilpotent restriction to each space E(o)X with a

An attempt to follow the development in the

**bounded**case by writing N = T —Sruns into difficulties. The

**operator**N, although easily seen by Lemma 2 to have aquasi-nilpotent restriction to each space E(o)X with a

**bounded**, need not be ...### 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 bounded spectral commuting compact complex numbers complex plane constant contains continuous functions converges Corollary countably additive Definition denote differential operator disjoint Doklady Akad domain eigenvalues elements equation equivalent exists finite number Foias follows from Lemma follows from Theorem formal differential operator formula Hence Hilbert space hypothesis identity inequality inverse Lebesgue Math matrix multiplicity Nauk SSSR norm operators in Hilbert perturbation polynomial preceding 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