## Linear Operators: Spectral operators |

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Page 2037

... D ( B ) , we have , by definition , lim î ( h ) - Î . - W = h Bils nn0 + On the other

hand , it follows from ( 55 ) that for each s in RN lim if ( h ) - 1 1 h = Â ( 8 ) { ( s ) , h

+ 0 + which shows that Â ( s ) + ( s ) is

.

... D ( B ) , we have , by definition , lim î ( h ) - Î . - W = h Bils nn0 + On the other

hand , it follows from ( 55 ) that for each s in RN lim if ( h ) - 1 1 h = Â ( 8 ) { ( s ) , h

+ 0 + which shows that Â ( s ) + ( s ) is

**square integrable**on RN and that Bys = ÂU.

Page 2332

... sum of

length at most 1 / B , and immediately apply the ordinary theory of orthogonal

expansions in Hilbert space ( cf . Theorem IV . 4 . 13 ) to obtain the desired result

.

... sum of

**square integrable**functions , each vanishing outside an interval oflength at most 1 / B , and immediately apply the ordinary theory of orthogonal

expansions in Hilbert space ( cf . Theorem IV . 4 . 13 ) to obtain the desired result

.

Page 2481

... x , are

functions ay , Qy , and a are defined in E " , bounded , and approach zero as [ 2 ]

→0 in E " . Let 11 = 3 , asko ) amewe , + acam bent ale ) . 1 = 1 Show that , for 1 <

0 ...

... x , are

**square integrable**for all 1 Si , j n . ( b ) Suppose that the coefficientfunctions ay , Qy , and a are defined in E " , bounded , and approach zero as [ 2 ]

→0 in E " . Let 11 = 3 , asko ) amewe , + acam bent ale ) . 1 = 1 Show that , for 1 <

0 ...

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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