(Adapted from [Rockafellar](http://www.math.washington.edu/~rtr/papers/rtr066-MonoOpProxPoint.pdf))



> 
> **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



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


