Suppose we are given a linear operator $L$ on a Banach space $X$. Is there any way to extend $L$ to a multi-linear operator $\mathcal{L}$ in such a way that $$\mathcal{L}(x_1, x_2^*, \ldots, x_n^*) = L(x_1)$$ for some fixed values of $x^*_2, \ldots, x_n^*$?

If this seems too difficult, any insight on how to canonically associate $L$ to a multi-linear operator would also be interesting.

`$L(x_1)f(x_2)\dots f(x_n)$`

where $f$ is some non-zero linear functional on $X$. Then take `$x_2^*=\dots=x_n^*=y$, where $y$ is chosen so that $f(y)=1$. – Andreas Blass Feb 13 '11 at 23:50