## 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 2203

Since o ( E ) ( F ) = 0 ( EF ) , the sets 0 ( E ) actually form a

in 1 ( cf . Theorem IX . 2 . 11 and its proof ) . Consider now a closed set 8 , with 8

58 , Se . Each point in d , is interior to some set o ( E ) şe . Since 8 , is compact , a

...

Since o ( E ) ( F ) = 0 ( EF ) , the sets 0 ( E ) actually form a

**basis**for the topologyin 1 ( cf . Theorem IX . 2 . 11 and its proof ) . Consider now a closed set 8 , with 8

58 , Se . Each point in d , is interior to some set o ( E ) şe . Since 8 , is compact , a

...

Page 2313

None of this should cause the reader any essential difficulty . 9 LEMMA . Let E be

a projection of a B - space X onto a finite dimensional range and let E * : X * →X *

be its adjoint . Then , if 91 , 92 , . . . , Pn is a

None of this should cause the reader any essential difficulty . 9 LEMMA . Let E be

a projection of a B - space X onto a finite dimensional range and let E * : X * →X *

be its adjoint . Then , if 91 , 92 , . . . , Pn is a

**basis**for EX , we can find a unique ...Page 2451

... Ln = 1 n = 1 n = ) = | | RA | | 2 = | | A | | | 2 Put ( 13 ) T ( A ) ( 2m , A * xn ) , = 2 ( x ,

xm ) xn n , m = 1 dm - dm mm for x e X and A A . By Schwarz ' inequality , and

since { xn } is an orthonormal

2 ...

... Ln = 1 n = 1 n = ) = | | RA | | 2 = | | A | | | 2 Put ( 13 ) T ( A ) ( 2m , A * xn ) , = 2 ( x ,

xm ) xn n , m = 1 dm - dm mm for x e X and A A . By Schwarz ' inequality , and

since { xn } is an orthonormal

**basis**, we have n = 1 m = 1 men Ern ? | ( Axm , Xn )2 ...

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### Contents

SPECTRAL OPERATORS XV Spectral Operators | 1924 |

Introduction | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

32 other sections not shown

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analytic apply arbitrary assumed B-space Banach space Boolean algebra Borel sets boundary conditions bounded bounded Borel bounded operator Chapter clear clearly closure commuting compact complex consider constant contained converges Corollary corresponding countably additive defined Definition denote dense determined differential operator 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 manifold Math Moreover multiplicity norm positive preceding present problem projections PROOF properties prove range regular resolution resolvent respectively restriction Russian satisfies scalar type seen sequence shown shows spectral measure spectral operator spectrum statement strongly subset subspace sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector weakly zero