Partial results on composition of operators such that overall composition is monotone

Question

(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

Answer 1

For $0\le \theta\le\pi/2$, say that $T:H\to H$ is $\theta$-monotone (therefore monotone) iff for all $x$, $y$ in $H$, $(x-y,Tx-Ty)\ge \|x-y\|\,\|Tx-Ty\|\cos\theta$, that is, $x-y$ and $Tx-Ty$ make an angle not larger than $\theta$. Then, of course, if $T_i$ is a $\theta_i$-monotone operator for $i=0,\dots,1$ with $\theta_1+\theta_2\le \pi/2$, the composition $T_2T_1$ is $(\theta_1+\theta_2)$-monotone