An bouneded linear operator $A \in L(X, Y)$ (here $X$, $Y$ are Banach spaces) is called absolutely $2$-summable if there exists a $C>0$ such that $$ \left( \sum_{j=1}^N \| A x_j\|_X^2 \right)^{1/2} \leq C \cdot \sup \left\{ \left(\sum | \langle x_j, \omega \rangle | \right)^{1/2} \mid \omega \in X', \|\omega\|_{X'} = 1 \right\}$$ for any finite set of vectors $x_1, \dots x_N \in X$. The $2$-summable norm is the infimum of all such constants $C$.

Those operators are the natural analog to Hilbert-Schmidt-Operators: If $X$ and $Y$ are Hilbert spaces, than $A$ is absolutely $2$-summable if and only if $A$ is Hilbert-Schmidt, and the norms coincide.

**Now the question is:** If $X$ is a Hilbert space (but $Y$ isn't), can we restrict to elements $x_j$ of some orthonormal basis? That is, if we allow only elements of a given ONB to take for $x_1, \dots, x_N$, can the resulting $C$ be strictly smaller than the $2$-summable norm?