Operations with monotone operators and the monotonicity of the resulting operators (Q314493)

Let \(\mathcal H\) be a Hilbert space \({\mathcal D} \subseteq {\mathcal H}, \eta \in (-1,1)\) and \(T: {\mathcal D} \rightarrow {\mathcal H}\) be given. The authors say \(T\) to be \textit{\(\eta\)-increasing} if \(\langle Tx-Ty, x-y \rangle \geq \eta \parallel Tx-Ty \parallel \parallel x-y \parallel\) for all \(x,y \in {\mathcal D}\). For \(T\) to be \textit{\(\eta\)-decreasing}, the reverse inequality should hold. Such a scalar \(\eta\) is referred to as a \textit{parameter of monotonicity} of the operator \(T\). Moreover, \(T\) is said to be \textit{\(h\)-monotone} if \(|\langle Tx-Ty, x-y \rangle |< \| Tx-Ty \| \| x-y \|\) for all \(x,y \in {\mathcal D}\). Recall that \(T\) is called Minty-Browder monotone if \(\langle Tx-Ty, x-y \rangle \geq 0\) for all \(x,y \in {\mathcal D}\). In the paper under review, the authors obtain estimates for the parameters of monotonicity of the composition of two operators and for their sum. A global injectivity result is proved as an application of the obtained estimates. Examples of \(h\)-monotone operators which are not Minty-Browder monotone are given.