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2Midterm1 - Analysis Midterm 1 Write solutions in a neat...

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Analysis Midterm 1 Write solutions in a neat and logical fashion, giving complete reasons for all steps. 1. Let H be a complex Hilbert space. (a) Prove that the set F ( H ) comprising all finite-rank operators on H is a minimal ideal in B ( H ). (b) Prove that the set K ( H ) comprising all compact operators on H is a minimal closed ideal in B ( H ). Remark : In both parts, the indefinite article may be replaced by a definite. Solution We shall not spell out the details that F ( H ) and K ( H ) are ideals in B ( H ) as these are standard. (a) Let J be a nonzero ideal in B ( H ). Choose a nonzero operator A J ; we may choose ξ H so that η := is nonzero. Let x, y H be arbitrary and choose T B ( H ) so that = y ; for example, T = η - 2 η y works. Now compute: if v H then TA ( x ξ )( v ) = T ( A x | v ξ ) = x | v TAξ = x | v Tη = x | v y = ( x y )( v ) which places x y = TA ( x ξ ) in the ideal J . This shows that J contains each rank-one operator; finally, any finite-rank operator is the sum of finitely many rank-one operators.
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