On Banach spaces with bases
Let \(X\) be a separable Banach space and let \((U_n)\) be a sequence of linear operators \(U_n: X\to X\). \((U_n)\) factors uniformly \(\ell^{m_n}_p\)'s with respect to \(\lambda\), if there exists a sequence of positive integers \((m_n)\) and sequences of linear operators \((T_n)\), \((S_n)\), with \(T_n:X\to\ell^{m_n}_p\), \(S_n:\ell^{m_n}_p\to X\) such that \(U_n= S_nT_n\), \(|T_n|\leq\lambda\), \(|S_n|\leq\lambda\) for \(n=1,2,\dots\)\ . Let \(R_n:X\to X\) be bounded linear operators of finite rank. \((R_n)\) is called a commuting approximating sequence (c.a.s.), if \(\lim_{n\to\infty} R_nx=x\) for all \(x\in X\) and \(R_nR_m= R_{\min(n,m)}\) for \(n\neq m\). The main theorem states that if \(X\) has a c.a.s. \((R_n)\) such that \((R_n- R_{n-1})\) factors uniformly through \(\ell^{m_n}_p\)'s for some \(p\) \((1\leq p\leq\infty)\), then \(X\) has a basis.
scientific article; zbMATH DE number 908032