## 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 f

and g are both

commute and ...

This shows that ( vi ) holds for every

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

and g are both

**bounded**Borel functions . Thus the**operators**f ( T ) and g ( T )commute and ...

Page 2239

Since f xe is a bounded function , the operator T ( f Xe ) is a

x is in 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 ( ē ) x = T ( $ Xee )

...

Since f xe is a bounded function , the operator T ( f Xe ) is a

**bounded operator**. Ifx is in 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 ( ē ) x = T ( $ Xee )

...

Page 2252

An attempt to follow the development in the

runs into difficulties . The

quasi - nilpotent restriction to each space E ( 0 ) X with o

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 ( 0 ) X with o

**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 analytic applications arbitrary assumed B-space Banach space belongs Boolean algebra Borel sets boundary bounded bounded operator Chapter clear closed commuting compact complex condition consider constant contained continuous converges Corollary corresponding countably additive defined Definition denote dense determined domain elements equation equivalent established example exists extension fact finite follows formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator manifold Math Moreover multiplicity norm normal perturbation plane 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 uniformly unique valued vector weakly zero