mKdV Hamiltonian operators, the recursion operator in the mCH hierarchy has a simple relationship to the mKdV recursion operator . First, the split operators can be expressed in terms of the mCH operators by $$\H=\Delta\inv\wtil\H,
\quad
\E_1 = (1-\Delta) \H, 
\quad
\E_2 =\wtil\E .$$ Next, substitution of these operator expressions into the mKdV recursion operator yields $$\label{mkdv-Rrel1}
\R=\E\H\inv= \E_1\H\inv +\E_2\H\inv$$ where $$\label{mkdv-Rrel2}
\E_1\H\inv = 1-\Delta, 
\quad
\wtil\E_2\H\inv = \wtil\E_2\wtil\H\inv\Delta = \wtil\R\Delta .$$ Then, when these two equations – are combined, this gives the relation $$\label{mkdv-recursion-rel}
\R-1 =(\wtil\R-1)\Delta .$$

Since $\wtil\R$ is a hereditary recursion operator, so is $\wtil\R-1$, and therefore $$\label{mch-shifted-Rop}
\what\R = (\R-1)\Delta\inv$$ defines a hereditary recursion operator. The hereditary property of this operator can be verified to hold directly, using the hereditary property of the mKdV recursion operator $\R$, which does not require splitting of the mKdV Hamiltonian operators.

Furthermore, since $\R$ is invariant under $x$-translations, so is $\what\R$. Therefore, by a standard result in the theory of recursion operators [@Olv77; @Olv-book], $\what\R$ can be used to generate an integrable hierarchy of flows $\what P^{(n)} = \what\R^n(-v_x)$, $n=1,2,\ldots$, starting from the root flow $P^{(0)}=-v_x$ for the mKdV and mCH hierarchies. Note the integrability structure provided by the recursion 