## Linear Operators: Part III: Spectral Operators [by] Nelson Dunford and Jacob T. Schwartz, with the Assistance of William G. Bade and Robert G. Bartle, Volume 1 |

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Page 1979

It follows from equations ( iv ) and ( v ) of Lemma 3 that E ( dis ( s ) ; Â ( 8 ) ) is e - essentially bounded on S. Lemma 4 then

It follows from equations ( iv ) and ( v ) of Lemma 3 that E ( dis ( s ) ; Â ( 8 ) ) is e - essentially bounded on S. Lemma 4 then

**shows**that condition ( i ) of the theorem is satisfied . Q.E.D. 8 COROLLARY .Page 2169

This

This

**shows**that ( vi ) holds for every bounded Borel function f and every continuous 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 ) ...Page 2170

These lemmas will

These lemmas will

**show**that the hypotheses of Theorem 5.18 are satisfied by a self adjoint operator in Hilbert space . ... If a is not real , an expansion of the scalar product ( ( al - T ) x , ( al - T ' ) x )**shows**that | ( al — T ...### What people are saying - Write a review

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

47 other sections not shown

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### Common terms and phrases

adjoint operator Amer analytic apply arbitrary assumed B-space Banach space belongs Boolean algebra Borel set boundary conditions bounded bounded operator Chapter clear closed commuting compact complex constant contains continuous converges Corollary corresponding defined Definition denote dense determined differential operator domain elements equation equivalent established exists extension finite follows formal formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear operator Math Moreover multiplicity norm perturbation plane positive preceding present problem projections PROOF properties prove range resolution resolvent restriction Russian satisfies scalar type seen sequence shown shows similar solution spectral measure spectral operator spectrum subset sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector zero