Some results on the relationship between two-dimensional behaviors decompositions and the factor skew-primeness property
The aim of the paper is to extend the autonomous/controllable decomposition for linear shift-invariant complete 2D behaviors to a more general setting: given a 2D complete behavior \(\mathfrak{B}\) and one of its sub-behaviors \(\mathfrak{B}_{1}\), some conditions are emphasized under which a complete behavior \(\mathfrak{B}_{2}\) can be found such that \(\mathfrak{B}=\mathfrak{B}_{1}+\mathfrak{B}_{2}\) and \(\mathfrak{B}_{1}\cap\mathfrak{B}_{2}\) is finite dimensional and autonomous. The starting point is represented by a previous paper of the authors in which necessary and sufficient conditions for the more restrictive direct sum decomposition \(\mathfrak{B}=\mathfrak{B}_{1}\oplus\mathfrak{B}_{2}\) were established. The main result (Theorem 4.1) states one-way (necessary or sufficient) conditions for the decomposition \(\mathfrak{B}=\mathfrak{B}_{1}+\mathfrak{B}_{2}\) above, conditions which are related to the factor skew-primeness property of some suitable matrix pairs. Some counterexamples are used to show that these conditions are not two-way. Some special situations are emphasized in which these conditions are necessary and sufficient.
scientific article; zbMATH DE number 1788476