## Linear Operators: Spectral theory |

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Page 1083

Consequently , the series 4 ( 2 ) -18 ( 2 ) 1 = 4,2 " n = 2 of the preceding

converges in the Hilbert - Schmidt norm . 44 Let ( S , E , u ) be a positive measure

space . Then an operator A in the Hilbert space L2 ( S , E , u ) is of Hilbert ...

Consequently , the series 4 ( 2 ) -18 ( 2 ) 1 = 4,2 " n = 2 of the preceding

**exercise**converges in the Hilbert - Schmidt norm . 44 Let ( S , E , u ) be a positive measure

space . Then an operator A in the Hilbert space L2 ( S , E , u ) is of Hilbert ...

Page 1086

Show , finally , that by choosing As , 8 ) = 0 for all s in S , we obtain the result of

method of

of trace ...

Show , finally , that by choosing As , 8 ) = 0 for all s in S , we obtain the result of

**Exercise**46 as a special case of the present result . ( Hint : Generalize themethod of

**Exercise**46. ) 49 The operator A of Hilbert - Schmidt class is said to beof trace ...

Page 1087

( Hint : For ( d ) , use Weyl's inequality ,

50 ( Halberg ) Let ( S , E , u ) be a o - finite measure space . Let T , be a 1 -

parameter family of bounded operators defined in a subinterval I of the parameter

...

( Hint : For ( d ) , use Weyl's inequality ,

**Exercise**30. ) E. Miscellaneous**Exercises**50 ( Halberg ) Let ( S , E , u ) be a o - finite measure space . Let T , be a 1 -

parameter family of bounded operators defined in a subinterval I of the parameter

...

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