Kronecker product covariance structure provides an efficient way to modeling the inter-correlations of matrix-variate data. In this paper, we propose testing statistics for Kronecker product covariance matrix based on linear spectral statistics of renormalized sample covariance matrices. Central limit theorem is proved for the linear spectral statistics with explicit formulas for mean and covariance functions, which fills in the gap in the literature. We then theoretically justify that the proposed testing statistics have well-controlled sizes and strong powers. To facilitate practical usefulness, we further propose a bootstrap resampling algorithm to approximate the limiting distribution of associated linear spectral statistics. Consistency of the bootstrap procedure is guaranteed under mild conditions. Extensive numerical studies demonstrate empirically reliable performance of the proposed testing procedure. The approach is then applied to the analysis of a real data set with well-structured portfolio returns. Various estimation approaches lead to contradictory investing strategies in this example, while our testing procedure provides a guideline of selecting the most convincing strategy. This is joint with Yu Long and Xie Jiahui.