In single-channel speech enhancement, methods based on full-band spectral features have been widely studied. However, only a few methods pay attention to non-full-band spectral features. In this paper, we explore a knowledge distillation framework based on sub-band spectral mapping for single-channel speech enhancement. Specifically, we divide the full frequency band into multiple sub-bands and pre-train an elite-level sub-band enhancement model (teacher model) for each sub-band. These teacher models are dedicated to processing their own sub-bands. Next, under the teacher models' guidance, we train a general sub-band enhancement model (student model) that works for all sub-bands. Without increasing the number of model parameters and computational complexity, the student model's performance is further improved. To evaluate our proposed method, we conducted a large number of experiments on an open-source data set. The final experimental results show that the guidance from the elite-level teacher models dramatically improves the student model's performance, which exceeds the full-band model by employing fewer parameters.