## Linear Operators: Spectral operators |

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Page 2203

The mapping E H0 ( E ) is clearly an isomorphism between B and the Boolean

algebra of all open and closed subsets of 4 . These open and closed sets o ( E )

form a

The mapping E H0 ( E ) is clearly an isomorphism between B and the Boolean

algebra of all open and closed subsets of 4 . These open and closed sets o ( E )

form a

**basis**for the topology in 1 . To see this , note that sets of the form { a | | ( T ...Page 2451

A * Xn ) ( x , xnxn n . m = 1 in im n , m = 1 num for xe X and A € A . By Schwarz '

inequality , and since { xn } is an ortho . normal

3 : 14 : 46 16 . - ) ) si ( * 4 , - 1 ) ( is - > r " ) Slale , 72 | ( 48m , toal ? n = 1 m ...

A * Xn ) ( x , xnxn n . m = 1 in im n , m = 1 num for xe X and A € A . By Schwarz '

inequality , and since { xn } is an ortho . normal

**basis**, we have n = 1 min ( 14 ) ] (3 : 14 : 46 16 . - ) ) si ( * 4 , - 1 ) ( is - > r " ) Slale , 72 | ( 48m , toal ? n = 1 m ...

Page 2503

In Kato [ 14 ] this result is proved rigorously on the

hypotheses . Related results are given in Birman [ 3 ] . In this paper , theorems of

the nature of the wave operator invariance theorem are used to establish the

existence ...

In Kato [ 14 ] this result is proved rigorously on the

**basis**of rather generalhypotheses . Related results are given in Birman [ 3 ] . In this paper , theorems of

the nature of the wave operator invariance theorem are used to establish the

existence ...

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### Contents

SPECTRAL OPERATORS | 1924 |

Spectral Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

31 other sections not shown

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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