Let $H$ be a real Hilbert space with inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$, and induced norm $\left\| \cdot \right\|: H \rightarrow \mathbb{R}_{\geq 0}$.

Let $A, B : H \rightarrow H$ be monotone operators, that is (for both $A$ and $B$) $$ \langle A x - Ay, x-y\rangle \geq 0 \quad \forall x,y \in H$$

I am wondering if $A \circ B$ is monotone as well, that is, if $$ \langle A B x - A B y, x-y\rangle \geq 0 \quad \forall x,y \in H$$


The answer is no. Consider the linear operator $A x = \begin{bmatrix} 0 & -1\\ 1 & 0 \end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}$. Observe that it is monotone and consider $B=A$.

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