A near-factorization theorem for integrable functions (Q1316484)

Let \((Z,{\mathcal B},\mu)\) be a measure space, \(D\) a separable Hilbert space and \(H\) be a closed subspace of \(L^ 2 (\mu; D)\). Given \(x,y\in L^ 2 (\mu;D)\) define \(x\cdot y\in L^ 1(\mu)\) by \((x\cdot y) (\xi)= \langle x(\xi), y(\xi)\rangle\) for almost every \(\xi\in Z\). The paper under review is devoted to the proof of the following theorem. Suppose that \(L^ 2 (\mu; D)\) is separable. Assume that for every set \(\omega\in{\mathcal B}\) with positive measure, every \(\varepsilon>0\) and every set of vectors \(\xi_ 1,\dots, \xi_ p\in L^ 2(\mu; D)\), there exists \(z\in H\) such that \(| \Xi_{Z\setminus\omega} z|< \varepsilon | \Xi_ \omega z|\) and \(\langle z,\xi_ j\rangle_{L^ 2 (\mu; D)} =0\), \(j=1,\dots,p\). Then for every \(f\in L^ 1 (\mu)\), every \(\varepsilon>0\) and every set \(\xi_ 1,\dots, \xi_ p\in L^ 2 (\mu;D)\) there exist \(x,y\in H\) such that \(x,y\) are orthogonal to every \(\xi_ j\), their \(L^ 2 (\mu; D)\)-norm is less than the square root of the \(L^ 1\)-norm of \(f\), the \(L^ 1\)-norm of \(f- x\cdot y\) is less than \(\varepsilon\) and if \(f\geq 0\) almost everywhere then \(x=y\). This result had been proved by the first author [``Factorization theorems for integrable functions'', in ``Analysis at Urbana. II'', Lond. Math. Soc. Lect. Note Ser. 138, 9-21 (1989; Zbl 0681.46034)] without the assumption of separability but with the additional condition of \(\mu\)- essential boundedness for \(z\).