In multiple antenna systems employing time-division duplexing, spatial precoder design at the base station (BS) leverages channel state information acquired through uplink pilot transmission, under the assumption of channel reciprocity. Malicious eavesdroppers can start pilot spoofing attacks to alter such design, so as to improve their eavesdropping performance in downlink. The aim of this paper is to study the effects of pilot spoofing attacks on uplink channel estimation, by assuming that the BS knows the angle of arrivals (AoAs) of the legitimate channels. Specifically, after assessing the performance of the simple least squares estimator (LSE), we consider more sophisticated estimators, such as the maximum likelihood estimator (MLE) and different versions of the minimum mean square error estimator (MMSEE), involving different degrees of a priori information about the pilot spoofing attacks. Theoretical analysis and numerical simulations are used to compare the performance of such estimators. In particular, we analytically demonstrate that the spoofing effects in the high signal-to-noise regime can be completely suppressed, under certain conditions involving the AoAs of the legitimate and spoofing channels. Moreover, we show that even an imperfect knowledge of the AoAs and of the average transmission power of the spoofing signals allows the MLE and MMSEE to achieve significant performance gains over the LSE.