The likelihood ratio test for a separable covariance matrix

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Tests for separability in nonparametric covariance operators of random surfaces (2015, background): “Notation. Let us first introduce some definitions and notation about operators in a Hilbert space [see, e.g., Gohberg, Goldberg and Kaashoek (1990), Kadison and Ringrose (1997), Ringrose (1971)].”
Models with a Kronecker product covariance structure: Estimation and testing (2008, methodology): “Similarly to testing the hypothesis that the Kronecker product structure holds for the model against the general alternative, the restriction (1.4) is essential for obtaining the MLE unlike the one obtained by Lu and Zimmerman [ 4 ] and Roy and Khattree [8].”
Matrix variate regressions and envelope models (2016, background): “Hypothesis tests for the Kronecker covariance structure can be found in Shitan and Brockwell (1995), Lu and Zimmerman (2005) and Roy and Khattree (2005).”
Estimating and Testing a Structured Covariance Matrix for Three-Level Multivariate Data (2011, background): “Hypothesis testing problem on doubly multivariate data or two-level multivariate data using separable covariance structure V ⊗ , where V (u× u positive definite) and (q × q positive definite) are unstructured variance–covariance matrices on both the multivariate levels (say, site and response variab”