How to prove that a preconditioner cannot be superlinear

In the general case of multilevel Toeplitz matrices, we recently proved that any multilevel circulant preconditioner is not superlinear ( a cluster it may provide cannot be proper). The proof was based on the concept of quasi-equimodular matrices, although this concept does not apply, for example, to the sine-transform matrices. In this paper, with a new concept of partially equimodular matrices, we cover all trigonometric matrix algebras widely used in the literature. We propose a technique for proving the non-superlinearity of certain frequently used preconditioners for some representative sample multilevel matrices. At the same time, we show that these preconditioners are, in a certain sense, the best among the sublinear preconditioners ( with only a general cluster) for multilevel Toeplitz matrices.