Range inclusion and factorization of operators on classical Banach spaces (Q1174844)

The authors study the following question: If \(Z\) is a Banach space and \(A\) and \(B\) are bounded linear operators on \(Z\) with \(\hbox {Range} A\subset\hbox {Range} B\), then does there exist a bounded linear operator \(C\) on \(Z\) such that \(A=BC\)? If \(Z\) is a Hilbert space, the positive answer is known. The authors show that if \(Z\) is one of the classical Banach spaces \(\ell_ p\) (\(1<p<\infty\), \(p\neq 2\)), \(L_ p\) \((1\leq p<\infty, p\neq 2)\), \(c_ 0\) or \(C(K)\) with \(K\) compact and metric, then the answer is negative.