Compact operators in the C ^∗ -algebra generated by a matrix weighted shift

Complex symmetry operators have notable applications in extension and dilation results, rank one perturbations of Jordan operators, matrix-valued inner functions and free interpolation theory in the disk and so on. While in the study of the complex symmetric operators, one of the problems one always encounters is when a C^∗-algebra singly generated contains no nonzero compact operators. In this paper, we answer this question recur to matrix weighted shift operators.