Milimeter wave (mmWave) band mobile communications can be a solution to the continuously increasing traffic demand in modern wireless systems. Even though mmWave bands are scarcely occupied, the design of a prospect transceiver should guarantee the efficient coexistence with the incumbent services in these bands. To that end, in this paper, two underlay cognitive transceiver designs are proposed that enable the mmWave spectrum access while controlling the interference to the incumbent users. MmWave systems usually require large antenna arrays to achieve satisfactory performance and thus, they cannot support fully digital transceiver designs due to high demands in hardware complexity and power consumption. Thus, in order to develop efficient solutions, the proposed approaches are based on a hybrid analog-digital pre-coding architecture. In such hybrid designs, the overall beamformer can be factorized in a low dimensional digital counterpart applied in the baseband and in an analog one applied in the RF domain. The first cognitive solution developed in this paper designs the cognitive hybrid pre-coder by maximizing the mutual information between its two ends subject to interference, power and hardware constraints related to the analog counterpart. The second solution aims at reduced complexity requirements and thus derives the hybrid pre-coder by minimizing the Frobenious norm of its difference to the optimal digital only one. A novel solution for the post-coder at the cognitive receiver part is further proposed here based on a hardware constrained Minimum Mean Square Error criterion. Simulations show that the performance of both the proposed hybrid approaches is very close to the one of the fully digital solution for typical wireless environments.