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The acquisition of accurate channel state information is critical for enhancing the transmission quality and energy efficiency of reconfigurable intelligent surface-aided millimeter wave systems with multiple-input multiple-output antenna arrays at the transceiver ends. Due to the cascaded doubly sparse channel property in angular and delay domains, it is difficult to acquire nonzero angular gains sampled by overcomplete dictionaries with low pilot overhead. In this work, a channel-estimation method based on the sparse Kalman filter (SKF) framework is proposed, in which a state-space model is established for a sparse complex angular gain vector in both static and dynamic scenarios, and a zero-padding block-diagonal pilot pattern is designed. A linear <inline-formula> <tex-math notation="LaTeX">$l_{1}$ </tex-math></inline-formula>-norm subdifferential pseudo-measurement equation is deduced. The on-grid angular gain vector, along with the corresponding support, is recovered by the sequential processing of multiple pilot tone observations. To enhance the <inline-formula> <tex-math notation="LaTeX">${l}_{1}$ </tex-math></inline-formula>-norm convergence and thus reduce the pilot overhead, an accelerated algorithm is developed, which incorporates an exponential factor to accelerate the gradient descent procedure. The simulation results demonstrate that the method can capture the angular support using a few pilot blocks. The normalized mean square error performance for the angular gain vector was better than that for the orthogonal matching pursuit. The accelerated SKF algorithm achieves faster convergence and higher accuracy compared with the ordinary SKF algorithm, and it is easy to implement online.