Characterization of the Gamma semigroup

Abstract

The Gamma semigroup with parameter $b>0$ on $L^p(\mathbb R^+)$ is defined by

$$\begin{aligned} W_b(t)f(x)=\frac{1}{\Gamma (t)}\int _0^x(x-y)^{t-1}e^{-b(x-y)}f(y)\,dy. \end{aligned}$$

Let S denote the multiplication operator $f(x)\rightarrow xf(x)$ with maximal domain D(S) in $L^p(\mathbb R^+)$. The bounded operator V on $L^p(\mathbb R^+)$ is S-Volterra if D(S) is V-invariant and $[S,V]=V^2$ on D(S). For $1<p<\infty$, we characterize the Gamma semigroup as the unique regular semigroup $V(\cdot )$ on $L^p(\mathbb R^+)$ with imaginary type less than $\pi$, such that V(1) is S-Volterra and $V(1)u^b=Su^b$, where $u^b(x):=e^{-bx}$.