Conformable Fractional Semigroups of Operators

Let $X$ be a Banach space, and $T:[0,\mathrm{\infty })\to \mathcal{L}(X,X),$ the bounded linear operators on $X\mathrm{.}$ A family ${T(t)}_{t\ge 0}\subseteq {% $\mathcal{L}$}(X,X)$iscalledaone-parametersemigroupif$T(s+t)=T(s)T(t),$and$T(0)=I,$theidentityoperatoron$X.$ The infinitesimal generator of the semigroup is the derivative of the semigroup at $t=0.$ The object of this paper is to introduce a (conformable) fractional semigroup of operators whose generator will be the fractional derivative of the semigroup at $t=0.$ The basic properties of such semigroups will be studied.