On the twisted factorization of the $T$-transform

The amalgamated $T$-transform of a non-commutative distribution was introduced by K.~Dykema. It provides a fundamental tool for computing distributions of random variables in Voiculescu’s free probability theory. The $T$-transform factorizes in a rather non-trivial way over a product of free random variables. In this article, we present a simple graphical proof of this property, followed by a more conceptual one, using the abstract setting of an operad with multiplication.