## Linear Operators: Spectral Theory : Self Adjoint Operators in Hilbert Space, Volume 2 |

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Page 2226

It was proved by Berkson [ 3 ] that if X is a

bounded Boolean algebra of projections on X , then X has an equivalent norm

such that the functionals given by Lemma 3.12 can be expressed by means of a ...

It was proved by Berkson [ 3 ] that if X is a

**uniformly**convex B - space and B is abounded Boolean algebra of projections on X , then X has an equivalent norm

such that the functionals given by Lemma 3.12 can be expressed by means of a ...

Page 2323

... S - 1 )

PROPERTIES OF ( ( 1/1 ) ( d / dx ) ) "

... S - 1 )

**uniformly**as Irl → 00 , je remaining in A XIX.4.1 2323 SPECTRALPROPERTIES OF ( ( 1/1 ) ( d / dx ) ) "

Page 2333

... m ( f ) is

by ( a ) , this is evident . Q.E.D. As remarked above , it follows immediately from

Lemma 7 that the collection of all finite sums of projections Em is bounded .

... m ( f ) is

**uniformly**bounded . Since { m } el , by hypothesis and { am ( f ) } is in l ,by ( a ) , this is evident . Q.E.D. As remarked above , it follows immediately from

Lemma 7 that the collection of all finite sums of projections Em is bounded .

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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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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 discrete domain elements equation equivalent established exists extension fact finite follows formal formula function 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