Exact factorization never holds for the Banach spaces of sequences on \(\mathbb{Z}\) (Q2761461)

Let \(\mathcal E\) denote the set of all complex sequences \(u=(u_n)_{n\in \mathbb{Z}}\) equipped with the topology of coordinatewise convergence and let \(S:(u_n)_{n\in \mathbb{Z}}\rightarrow (u_{n-1})_{n\in \mathbb{Z}}\) be the usual shift operator. For \(u\in\mathcal E\) we put \(\operatorname {supp}u=\{n\in \mathbb{Z}\mid u_n\neq 0\}\) and denote by \({\mathcal E}_0\) the set of sequences \(u\in\mathcal E\) such that \(\operatorname {supp}u\) is finite. We will say that a linear topological space \(\mathcal A\subset\mathcal E\) is admissible if the following three conditions are satisfied: NEWLINENEWLINENEWLINE(1) the injection \(i:\mathcal A\rightarrow\mathcal E\) is continuous; NEWLINENEWLINENEWLINE(2) \({\mathcal E}_0\subset\mathcal A\) and \({\mathcal E}_0\) is dense in \(\mathcal A\) with respect to the weak topology \(\sigma(\mathcal A, {\mathcal A}^*)\); NEWLINENEWLINENEWLINE(3) \(S(\mathcal A)=\mathcal A\) and the operators \(S:\mathcal A\rightarrow\mathcal A\) and \(S^{-1}:\mathcal A\rightarrow\mathcal A\) are continuous. NEWLINENEWLINENEWLINELet \({\mathcal A}^+=\{u\in{\mathcal A}\mid u_n=0\), \(n<0\}\), \({\mathcal A}^-=\{u\in{\mathcal A}\mid u_n=0\), \(n>0\}\). We denote by \({\mathcal L}({\mathcal A})\) the algebra of all continuous linear operators \(R:\mathcal A\rightarrow\mathcal A\), and by \({\mathcal M}({\mathcal A})\), \({\mathcal M}^+({\mathcal A})\), \({\mathcal M}^-({\mathcal A})\) the closure in \({\mathcal L}({\mathcal A})\) with respect to the strong operator topology of the sets \(\text{span}(S^n)_{n\in \mathbb{Z}}\), \(\operatorname {span}(S^n)_{n\leq 0}\), \(\operatorname {span}(S^n)_{n\geq 0}\) respectively. Let us say that an admissible linear topological space of sequences \(\mathcal A\) has the exact factorization property if for every \(u\in\mathcal A\) there exist \(k\geq 0\), \(R\in \text{Inv}({\mathcal M}^-({\mathcal A}))\) and \(v\in{\mathcal A}^+\) satisfying the condition \(u=S^{-k}Rv\). The main result of the paper under review is the following theorem. NEWLINENEWLINENEWLINETheorem 1.1. An admissible Banach space of sequences never has the exact factorization property. NEWLINENEWLINENEWLINEThe author gives some known examples of locally convex complete topological spaces of sequences that have the exact factorization property.