On thekth matrix numerical range

Let A be an n × n complex matrix. The kth matrix numerical range of A is the set W(k:A) = {X * AX:X is an n×k matrix such that X*X=Ik }. We study the conditions on A under which the set W(k:A) is convex or starshaped. We get complete answers for hermitian matrices and partial results for general matrices. Then we consider different matrix sets S such as the collection of all scalar matrices, hermitian matrices, normal matrices, etc., and investigate the following problems: does AεS imply XεS for all XεW(k:A); what can we say about A if all XεW(k:A) are elements of S; if what is the largest k such that Some possible variations of the problem and related results are also discussed.