Channel estimation is one of the main challenges for deploying the simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) in millimeter-wave (mmWave) systems, due to the large number of transmission and reflection channels. This letter proposes two channel estimation strategies for the STAR-RIS-assisted multi-user mmWave systems. Firstly, a sparse matrix recovery problem is proposed to estimate the cascaded channels, which comprise the channels from the base station to the STAR-RIS and from the STAR-RIS to the users. Subsequently, an algorithm based on the alternating direction method of multipliers (ADMM) is developed to exploit the sparsity and low-rank characteristics of mmWave channels, where the optimal solution of each variable can be computed analytically at each iteration. Additionally, a method based on parallel factor (PARAFAC) decomposition and the least-squares (LS) technique is proposed to estimate each channel separately. Finally, numerical results demonstrate the superiority of the proposed schemes over existing ones.