## Linear Operators: Spectral Theory : Self Adjoint Operators in Hilbert Space, Volume 2 |

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Page 2256

Since fo ( $ ) = 0 for S EC , D , and fo ( $ ) = 1 for Se Cu D , it follows immediately

from [ * ] that f . ( T ) 2pi . ( 51 – T ) -4a5 Q.E.D. 32 THEOREM . Let T be an

operator in a weakly complete B - space X.

disconnected .

Since fo ( $ ) = 0 for S EC , D , and fo ( $ ) = 1 for Se Cu D , it follows immediately

from [ * ] that f . ( T ) 2pi . ( 51 – T ) -4a5 Q.E.D. 32 THEOREM . Let T be an

operator in a weakly complete B - space X.

**Suppose**that o ( T ) is totallydisconnected .

Page 2284

However , rather than seek the maximum generality , it will be convenient to

Both these properties hold for B if X is separable , so this will be assumed for the

rest of ...

However , rather than seek the maximum generality , it will be convenient to

**suppose**that B is itself complete and satisfies the countable chain condition .Both these properties hold for B if X is separable , so this will be assumed for the

rest of ...

Page 2303

enumeration of its spectrum . Let dn denote the distance from in to o ( T ) - { n } .

.

**Suppose**that E is its resolution of the identity , and**suppose**that { an } is anenumeration of its spectrum . Let dn denote the distance from in to o ( T ) - { n } .

**Suppose**that for all but a finite number of n , E ( An ) has a onedimensional range.

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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 defined Definition denote dense determined differential operator discrete 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