## 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 2241

These equations show that T ( f ) and T ( 1 / f ) are one - to - one operators and

that the range of T ( f ) is a

in D ( T ( f ) ) , then x = T ( 1 / 8 ) T ( f ) x , and so x is in the range R ( T ( 1 / 8 ) ) of

T ...

These equations show that T ( f ) and T ( 1 / f ) are one - to - one operators and

that the range of T ( f ) is a

**subset**of the domain of T ( 1 / f ) and vice versa . If x isin D ( T ( f ) ) , then x = T ( 1 / 8 ) T ( f ) x , and so x is in the range R ( T ( 1 / 8 ) ) of

T ...

Page 2256

Since o ( T ) is totally disconnected , each point , in o ( T ) is contained in an

arbitrarily small compact

of o ( T ) . It follows that the set 7 ( 0 ) = { 212 - 1€o } is a compact

...

Since o ( T ) is totally disconnected , each point , in o ( T ) is contained in an

arbitrarily small compact

**subset**o of o ( T ) which is open in the relative topologyof o ( T ) . It follows that the set 7 ( 0 ) = { 212 - 1€o } is a compact

**subset**of o ( R )...

Page 2257

Since by assumption | E ( 0 ; T ) | is uniformly bounded for o in K , since each

compact open

is uniformly bounded as oy runs over the family of all compact open

...

Since by assumption | E ( 0 ; T ) | is uniformly bounded for o in K , since each

compact open

**subset**o , of o ( R ) is ... 10 , it follows immediately that E ( 01 ; R )is uniformly bounded as oy runs over the family of all compact open

**subsets**of o...

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### Contents

SPECTRAL OPERATORS | 1924 |

Introduction | 1927 |

Terminology and Preliminary Notions | 1929 |

Copyright | |

28 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 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