Let $\mathcal{L}_p(H)$ denote the space of compact operators on $H$ with singular values in $\ell^p$. In their article Noncommutative maximal ergodic theorems, Junge and Xu define the space $E$ as the set of all sequences $(x_n)$ in $\mathcal{L}_p(H)$ that admit a factorization $x_n=a y_n b$ with $a,b\in \mathcal{L}_{2p}(H)$ and $(y_n)$ a bounded sequence in $\mathcal{L}(H)$. This space is endowed with the norm $$ \|(x_n)\|_E=\inf\{\|a\|_{2p}\cdot\sup_{n}\|y_n\|\cdot\|b\|_{2p}: x_n=ay_n b\}. $$ It is stated that one can check that $E$ is a Banach space. I have already problems with the triangle inequality.
If $x_n=ay_n b$ and $\tilde x_n=\tilde a\tilde y_n \tilde b$, then one can apply Douglas' lemma to get contractions $c,\tilde c,d,\tilde d$ such that $$ x_n+\tilde x_n=(aa^\ast+\tilde a \tilde a^\ast)^{1/2}(c y_n d+\tilde c\tilde y_n \tilde d)(b^\ast b+\tilde b^\ast \tilde b)^{1/2}. $$ Clearly $(aa^\ast+\tilde a \tilde a^\ast)^{1/2},(b^\ast b+\tilde b^\ast \tilde b)^{1/2}\in \mathcal{L}_{2p}(H)$, so that $x_n+\tilde x_n\in E$. However, the naive road from here (triangle inequality and submultiplicativity) does not seem to yield the desired result.
Question: How to show that $\lVert\cdot\rVert_E$ is indeed a norm?