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.



Special kind of operators
See page 228ff of


Albrecht Pietsch: Operator ideals, Elsevier 1980.
[(pdf here)](http://gen.lib.rus.ec/book/index.php?md5=BAD44BB627E2D4148E9F042170671CDB)


Maybe, your operators are the $(\infty, p, \infty)$-summing operators there.


