On normed products of operator ideals which contain frak{L}₂ as a factor

We investigate quasi-Banach operator ideal products $({\frak{A}}\circ{\frak{B}},\mathbf{A\circ B})$ which contain $(\frak{L}_2, \mathbf{L}_2)$ as a factor. In particular, we ask for conditions which guarantee that $\mathbf{A\circ B}$ is even a norm if each factor of the product is a 1-Banach ideal. In doing so, we reveal the strong influence of the existence of such a norm in relation to the accessibility of the product ideal and the structure of its factors.