Let \(f\) be an operator-valued function, analytic on the open unit disc \(\mathbb{D}\), whose values act from a Banach space \(E\) into a Banach space \(F\). This paper concerns the general problem to factor \(f\) in the form \(f(z)= g(z) h(z)\), where \(g\) and \(h\) are again analytic on \(\mathbb{D}\), and the factorization is through a Hilbert space. More precisely, in an abstract setting, the paper deals with the following question. For which Banach spaces \(X\) and \(Y\) is it true that the natural product map \(Q_{X,Y}: H^ 2(X) \widehat {\otimes} H^ 2(Y)\to H^ 1(X \widehat {\otimes} Y)\) is surjective? Here \(X\widehat {\otimes}Y\) denotes the projective tensor product of \(X\) and \(Y\), and \(H^ p(X)\) is the usual Hardy space of \(X\)-valued analytic functions on the open unit disc \(\mathbb{D}\). An analogous question is considered for the map \(\widetilde{Q}_{X,Y}: \widetilde {H}^ 2(X) \widehat {\otimes} \widetilde {H}^ 2(Y)\to \widetilde {H}^ 1 (X\widehat {\otimes} Y)\), where \(\widetilde{H}^ p(X)\) is the closure in \(H^ p(X)\) of the polynomials with coefficients in \(X\), and the map \(\widetilde {Q}_{X,Y}\) is obtained by restricting \(Q_{X,Y}\) to polynomials. The author shows that the map \(\widetilde {Q}_{X,Y}\) is surjective whenever \(X\), \(Y\) are Banach spaces of type 2 or whenever \(X^*\), \(Y^*\) are of cotype 2 and satisfy some additional restrictive conditions. With a supplementary assumption on \(X\), \(Y\) (uniform \(H^ 2\)-convexity) it follows that \(Q_{X,Y}\) itself is surjective. Without additional restrictive conditions in the cotype 2 case surjectivity may fail to hold as is known from counterexamples constructed by \textit{O. Kouba} [C. R. Acad. Sci. Paris, Sér. I 307, No. 19, 949-953 (1988; Zbl 0662.46075)]. Special attention is paid to the case when \(X\), \(Y\) are Banach lattices; \(\widetilde {Q}_{X,Y}\) is shown to be surjective if \(X\), \(Y\) are both 2-convex Banach lattices. The proof of the author's main factorization theorem is based on the Sz-Nagy-Foiaş commutant lifting theorem which is explained in an appendix. In a second appendix the author shows how this main factorization theorem may be derived from the classical factorization of positive matrix functions.