{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

2Midterm1

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

This preview shows pages 1–2. Sign up to view the full content.

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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}