(Adapted from Rockafellar)

Definition: Let $H$ be a real Hilbert space with inner product $\langle \cdot ,\cdot \rangle$. A function $T: H \to H$ is said to be a monotone operator if \begin{equation} \langle z - z', Tz-Tz'\rangle \geq 0 \end{equation}

Let $A: H_1 \to H_2$, $B: H_2 \to H_3$ be monotone operators, then define their composition to be $B\circ A:H_1 \to H_3$. The composition $B \circ A$ is monotone if for all $z, z' \in H_1$

\begin{equation} \langle z - z', (B \circ A)z-(B \circ A)z'\rangle \geq 0 \end{equation}

It is well known that, in general, composition of monotone operators are not monotone.

Now suppose $A$ is monotone. Are there partial results in the literature that places condition on $B$, such that $B\circ A$ is monotone