ACTA issues

## Characterization of the Volterra operator and the Riemann-Liouville semigroup

Shmuel Kantorovitz

Acta Sci. Math. (Szeged) 80:3-4(2014), 459-466
70/2012

 Abstract. The Volterra operator $V\colon f(x)\to\int _0^xf(t) dt$ on $C[0,1]$ or $L^p(0,1)$ ($1\leq p< \infty$) is characterized as the unique bounded linear operator on the space satisfying the algebraic condition $(*)$ $[S,V]=V^2$, $Ve=Se$, where $S$ is the multiplication operator $f(x)\to xf(x)$, and $e$ is the function with constant value $1$. Similarly, if $S_\alpha$ is the multiplication operator $f\to\alpha f$ on $C[0,1]$, where $\alpha$ is a given injective real-valued $C[0,1]$-function of bounded variation vanishing at $0$, then the Stieltjes--Volterra operator $f(x)\to\int _0^xf(t) d\alpha(t)$ on $C[0,1]$ is characterized as the unique bounded linear operator on the space satisfying the above condition with $S=S_\alpha$. For $1< p< \infty$, the Riemann--Liouville semigroup is characterized as the unique regular semigroup $V(\cdot )$ on $\msbm C^+$ acting in $L^p(0,1)$, whose boundary group's type is less than $\pi$, and for which $V:=V(1)$ satisfies Relation~$(*)$. DOI: 10.14232/actasm-012-570-7 AMS Subject Classification (1991): 47D03, 26A42, 47G10, 97I50 Keyword(s): Volterra operator, Volterra Communication Relation, Stieltjes--Volterra operator, $C_0$-semigroup, regular semigroup, boundary group, type (of $C_0$-semigroup), Riemann--Liouville semigroup, Uniqueness theorem Received September 7, 2012, and in revised form February 14, 2013. (Registered under 70/2012.)