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

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Page 2256

It follows that the set 7 ( a ) = { z12-1 € 0 } is a compact subset of o ( R ) , open in

the relative topology of o ( R ) . Thus each point in o ( R ) different from

contained in an arbitrarily small compact set open in the relative topology of o ( R

) ...

It follows that the set 7 ( a ) = { z12-1 € 0 } is a compact subset of o ( R ) , open in

the relative topology of o ( R ) . Thus each point in o ( R ) different from

**zero**iscontained in an arbitrarily small compact set open in the relative topology of o ( R

) ...

Page 2325

0 3 Rz < 211 onto the w - plane with

our original observation ... In order to obtain information on the

from this , we now use Lemma 3 . We know by Lemma 2 that all but a finite

number ...

0 3 Rz < 211 onto the w - plane with

**zero**removed . Since sin z = ( 1 / 2i ) h ( et ) ,our original observation ... In order to obtain information on the

**zeros**of M ( u )from this , we now use Lemma 3 . We know by Lemma 2 that all but a finite

number ...

Page 2462

Moreover , if C belongs to the trace class 61 , then TnC converges to

norm , and CT * converges to

|| 2 < 1 } ) is conditionally compact , and thus for each ε > there exists a finite set ...

Moreover , if C belongs to the trace class 61 , then TnC converges to

**zero**in tracenorm , and CT * converges to

**zero**in trace norm . PROOF . The set K = C ( { Xe H|| 2 < 1 } ) is conditionally compact , and thus for each ε > there exists a finite set ...

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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