This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 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 finiterank 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 H so that := A is nonzero. Let x, y H be arbitrary and choose T B ( H ) so that T = y ; for example, T = k k 2 y works. Now compute: if v H then TA ( x )( v ) = T ( A h x  v i ) = h x  v i TA = h x  v i T = h x  v i y = ( x y )( v ) which places x y = TA ( x ) in the ideal J . This shows that J contains each rankone operator; finally, any finiterank operator is the sum of finitely...
View
Full
Document
 Fall '09
 Robinson

Click to edit the document details