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.



Extending linear operators to multi-linear ones
Following Kate's suggestion, I'm posting my previous comment as an answer. Choose any non-zero linear functional $f$ on $X$ and any vector $y\in X$ such that $f(y)=1$. Define $\mathcal L(x\_1,x\_2,\dots,x\_n)$  

 to be the product $L(x\_1)f(x\_2)\dots f(x\_n)$ and take all of 
$x\_2^\*,\dots x\_n^\*$ to be $y$.


