This paper focuses on multiuser MIMO channel estimation and data transmission at millimeter wave (mmWave) frequencies. The proposed approach relies on the time-division-duplex (TDD) protocol and is based on two distinct phases. First of all, the Base Station (BS) sends a suitable probing signal so that all the Mobile Stations (MSs), using a subspace tracking algorithm, can estimate the dominant left singular vectors of their BS-to-MS propagation channel. Then, each MS, using the estimated dominant left singular vectors as pre-coding beamformers, sends a suitable pilot sequence so that the BS can estimate the corresponding right dominant channel singular vectors and the corresponding eigenvalues. The low-complexity projection approximation subspace tracking with deflation (PASTd) algorithm is used at the MSs for dominant subspace estimation, while pilot-matched (PM) and zero-forcing (ZF) reception is used at the BS. The proposed algorithms can be used in conjuction with an analog RF beamformer and are shown to exhibit very good performance.