## 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 2021

18 I s

, 0 8€ 8° . ( 25 ) A - + 6 ) = ( . ) 0 + Se RP and ( 26 ) TÂ ( s ) ¥ ( 8 ) | = 18114 ( s ) ] ,

SE S , WE H2 . The

18 I s

**Equation**( 24 ) shows that | s \ - - Â ( s ) is unitary for 8 + 0 . Thus ( 40 ) * = 1, 0 8€ 8° . ( 25 ) A - + 6 ) = ( . ) 0 + Se RP and ( 26 ) TÂ ( s ) ¥ ( 8 ) | = 18114 ( s ) ] ,

SE S , WE H2 . The

**equation**Ap = 0 , being equivalent to the Cauchy - Riemann ...Page 2074

Now let y be an arbitrary vector in H . and define the vector x by the

) . Then ( 31 ) shows that x is in H . and

some vector 2 in H - we have e519 - by = e - Be - 569 + x + 2 , and , using ( 30 ) ...

Now let y be an arbitrary vector in H . and define the vector x by the

**equation**( 36) . Then ( 31 ) shows that x is in H . and

**equation**( 35 ) holds . This means that forsome vector 2 in H - we have e519 - by = e - Be - 569 + x + 2 , and , using ( 30 ) ...

Page 2401

( 2 ) ( 5 ) assuming that U has the form U = I + T ( B ) , with Be A . Taking U to be of

this form , we see that

( A2 ) ) = T ( I + T ( B ) ) , that is , to ( 3 ) T ( B ) T – TT ( B ) = - 1 ( B ) q ( A4 ) — 9 ...

( 2 ) ( 5 ) assuming that U has the form U = I + T ( B ) , with Be A . Taking U to be of

this form , we see that

**equation**( 1 ) is equivalent to the**equation**( I + T ( B ) ) ( T +( A2 ) ) = T ( I + T ( B ) ) , that is , to ( 3 ) T ( B ) T – TT ( B ) = - 1 ( B ) q ( A4 ) — 9 ...

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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 countably additive defined Definition denote dense determined differential operator 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