## Linear Operators: Spectral operators |

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Page 2084

Using the fact that the difference R ( A ; A ) – R ( A ; B ) is analytic for 2 # 0 , prove

that C is a quasi - nilpotent operator and that R ( A ; A ) = R ( A ; B ) + R ( A ; C ) –

55 ( McCarthy ) Let T be a spectral operator in a

Using the fact that the difference R ( A ; A ) – R ( A ; B ) is analytic for 2 # 0 , prove

that C is a quasi - nilpotent operator and that R ( A ; A ) = R ( A ; B ) + R ( A ; C ) –

55 ( McCarthy ) Let T be a spectral operator in a

**complex**B - space X which ...Page 2171

Exercises Some of the exercises will use the following notation . The symbol T is

a bounded linear operator on a

[ x ] will be used for the closed linear manifold determined by all the vectors R ( £

...

Exercises Some of the exercises will use the following notation . The symbol T is

a bounded linear operator on a

**complex**B - space X . For each x in X the symbol[ x ] will be used for the closed linear manifold determined by all the vectors R ( £

...

Page 2188

Let E be a spectral measure in the

countably additive on a o - field of subsets of a set 1 and let g be a bounded Borel

measurable function defined on the

...

Let E be a spectral measure in the

**complex**B - space X which is defined andcountably additive on a o - field of subsets of a set 1 and let g be a bounded Borel

measurable function defined on the

**complex**plane . Then $ 9 ( f ( n ) ) E ( da ) = g...

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### Contents

SPECTRAL OPERATORS | 1924 |

An Operational Calculus for Bounded Spectral | 1941 |

Part | 1950 |

Copyright | |

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