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

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Page 2070

... space under the

We let A , consist of all operators in H which have the form ( 17 ) a = ae + f , fe L.

Theorem 1 shows that the operator a determines a and f uniquely , and so we

may ...

... space under the

**norm**( 16 ) Iflo = \ f11 + llfll , felo , where | f || is the**norm**in X.We let A , consist of all operators in H which have the form ( 17 ) a = ae + f , fe L.

Theorem 1 shows that the operator a determines a and f uniquely , and so we

may ...

Page 2450

Let A be the set of all operators A € B ( X ) for which AX & D ( R ) and for which

RA belongs to the HilbertSchmidt class HS , and let ||| A || || RA || , so that the

Let A be the set of all operators A € B ( X ) for which AX & D ( R ) and for which

RA belongs to the HilbertSchmidt class HS , and let ||| A || || RA || , so that the

**norm**of an element A ¢ A is simply the Hilbert - Schmidt**norm**of the operator RA .Page 2462

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

|| 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 trace

**norm**, 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