A matrix nearness problem related to iterative methods

We consider a matrix nearness problem arising from an analysis of the speed of convergence of GMREs for solving a linear system Ax = b with A is an element of C-nxn and b is an element of C-n. More precisely, denoting by F-k the set of matrices of rank k at most, we solve [GRAPHICS] where S subset of C-nxn denotes the set of matrices of the form e(i theta) H - lambdaI with theta is an element of [0, 2 pi), lambda is an element of C, and H belonging to the set of Hermitian matrices. As to iterative methods, the set S is of interest in a larger context. To give an example, having a regular splitting A = S + F-k of A, the system Ax = b can be solved using a (k+3)-term recurrence with inner-outer iterations.