Asymptotic equivalence of a subclass of WCLT non degenerate GKSL generator

We prove the assymptotic equivalence of a sequence of block diagonal matrices with Toeplitz blocks. The blocks are the principal submatrices of an originating Toeplitz sequence with generating symbol of the Wiener class. As an application, using the invariance of certain $\mathit{\text{diagonal}}$ and $\mathit{\text{cyclic-diagonal}}$ operator subspaces of the GKSL generators of circulant and WCLT quantum Markov semigroups, the asymptotic equivalence of the families is proved under suitable hypothesis.