Hybrid beamforming (HBF) is an attractive technology for practical massive multiple-input and multiple-output (MIMO) millimeter wave (mmWave) systems. Compared with the fully-connected HBF architecture, the partially-connected one can further reduce the hardware cost and power consumption. However, the special block diagonal structure of its analog beamforming matrix brings additional design challenges. In this paper, we develop effective HBF algorithms for spectral efficiency maximization (SEM) in mmWave massive MIMO systems with the partially-connected architecture. One main contribution is that we prove the equivalence of the SEM problem and a matrix weighted sum mean square error minimization (WMMSE) problem, which leads to a convenient algorithmic approach to directly tackle the SEM problem. Specifically, we decompose the equivalent WMMSE problem into the hybrid precoding and hybrid combining subproblems, for which both the optimal digital precoder and combiner have closed-form solutions. For the more challenging analog precoder and combiner, we propose an element iteration based algorithm and a manifold optimization based algorithm. Finally, the hybrid precoder and combiner are alternatively updated. The overall HBF algorithms are proved to monotonously increase the spectral efficiency and converge. Furthermore, we also propose modified algorithms with reduced computational complexity and finite-resolution phase shifters. Simulation results demonstrate that the proposed HBF algorithms achieve significant performance gains over conventional algorithms.