## Linear Operators: Spectral operators |

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Page 2070

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

may define a

that A , is a B - space under this

...

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

may define a

**norm**in A , by the equation ( 18 ) lalo = 10 + iflo , ae A . . It is clearthat A , is a B - space under this

**norm**. It is also an algebra , for the product of two...

Page 2450

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

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

the

RA ...

Let A be the set of all operators A e B ( X ) for which AX Ç 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 operatorRA ...

Page 2462

Moreover , if C belongs to the trace class C1 , then TnC converges to zero in

trace

XE H | | x 3 1 } ) is conditionally compact , and thus for each a > 0 there exists a

finite ...

Moreover , if C belongs to the trace class C1 , then TnC converges to zero in

trace

**norm**, and CT * converges to zero in trace**norm**. PROOF . The set K = C ( {XE H | | x 3 1 } ) is conditionally compact , and thus for each a > 0 there exists a

finite ...

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### Contents

SPECTRAL OPERATORS | 1924 |

Spectral Operators | 1925 |

Terminology and Preliminary Notions | 1928 |

Copyright | |

31 other sections not shown

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adjoint operator analytic applications arbitrary assumed B-space Banach space belongs Boolean algebra Borel sets boundary bounded bounded operator Chapter clear closed commuting compact complex condition consider constant contained continuous converges Corollary corresponding countably additive defined Definition denote dense determined domain elements equation equivalent established example exists extension fact finite follows formula given gives Hence Hilbert space hypothesis identity inequality integral invariant inverse Lemma limit linear linear operator manifold Math Moreover multiplicity norm normal perturbation plane positive preceding present problem projections PROOF properties proved range regular resolution resolvent respectively restriction Russian satisfies scalar type seen sequence shown shows similar spectral measure spectral operator spectrum subset subspace sufficiently Suppose Theorem theory topology unbounded uniformly unique valued vector weakly zero