Consider an operator $A: H \longrightarrow X$ ($H$ is a Hilbert space and $X$ is a Banach space) that has a representation $$ A = \sum_{j=0}^\infty a_j \langle \cdot, e_j\rangle \cdot x_j,$$ where $(e_j)$ is an orthonormal basis of $H$ and $(x_j)$ is a family of vectors in $X$ with $\|x_j\| = 1$ and $(a_j) \in \ell_p(\mathbb{N})$.

Is there a special name for such operators? For a while I thought that these were just the absolutely $2$-summing operators between $H$ and $X$, but this seems to be wrong.

To give some background, if we have such an operator with $p=2$ and a bounded bilinear form $L$ on $X$, then the bilinear form $M$ on $H$ defined by $$M(v, w) = L(Av, Aw)$$ is trace-class, which I am interested in.