THE ABSTRACT CAUCHY PROBLEM AND A GENERALIZATION OF THE LUMER-PHILLIPS THEOREM

For injective, bounded operator $C$ on a Banach space $X$, the author defines the $C$-dissipative operator, and then gives Lumer-Phillips characterizations of the generators of quasi-contractive $C$-semigroups, where a $C$-semigroup $T(\cdot)$ is quasi-contractive if $\|T(t)x\|\le \|Cx\|$ for all $t\ge 0$ and $x\in X$. This kind of generators guarantee that the associate abstract Cauchy problem $u'(t,x)=Au(t,x)$ has a unique nonincreasing solution when the initial data is in $C(D(A))$ (here $D(A)$ is the domain of $A$). Also, the generators of quasi-isometric $C$-semigroups are characterized