On star-centers of some generalized numerical ranges and diagonals of normal matrices (Q5954127)

For each \(n\times n\) matrix \(A\) over the complex numbers three sets are considered. Firstly, the \(C\)-numerical range \(W_C(A)\) of \(A\), i.e. the set of all complex numbers tr\((CU^{*}AU)\), where \(U\) ranges in the unitary group \({\mathbf U}_{n}\) and \(C\) is a fixed complex \(n\times n\) matrix. Secondly, the set \(\operatorname {diag}{\mathbf U}(A)\) of all diagonal matrices that arise as follows: For each matrix \(U^{*}AU\) in the unitary orbit \({\mathbf U}(A)\) of \(A\) all the off-diagonal entries are replaced with \(0\). Thirdly, the set \({\mathbf S}(A)\) of all matrices whose \(C\)-numerical ranges are contained in \(W_C(A)\) for all matrices \(C\). NEWLINENEWLINENEWLINEThe major topic of the paper is to investigate the convexity of these sets or, more generally, their star-shapedness, and to determine the set of all star-centers. If \(A\) is normal then the set of star-centers of \(W_{A^*}(A)\) is a bounded closed real interval; also the sets of star-centers of \(\operatorname {diag}{\mathbf U}(A)\) and \({\mathbf S}(A)\) are described for a normal matrix \(A\). Furthermore, if \(A\) is normal and if its eigenvalues are non-collinear in the plane of complex numbers, then each of \(\operatorname {diag}{\mathbf U}(A)\) and \({\mathbf S}(A)\) has a unique star-center.