M-dissipative-operator

Let $X$ be a real Banach space with dual space $X ^ { * }$ and normalized duality mapping $J$ (cf. also Duality; Adjoint space). An operator $T : X \supset D ( T ) \rightarrow 2 ^ { X }$ is called dissipative if for every $x , y \in D ( T )$ and every $u \in T x , v \in T y$ there exists a $j \in J ( x - y )$ such that $\langle u - v , j \rangle \leq 0$ (cf. also Dissipative operator). A dissipative operator $T$ is called $m$-dissipative if $\lambda I - T$ is surjective for all $\lambda > 0$. Thus, an operator $T$ is dissipative (respectively, $m$-dissipative) if and only if the operator $- T$ is accretive (respectively, $m$-accretive). For more information, see Accretive mapping and $m$-accretive operator.