## Abstract

The Volterra operator V : f(x) → R ^{x} _{0} f(t) dt on C[0, 1] or L^{p}(0, 1) (1 ≤ p < ∞) is characterized as the unique bounded linear operator on the space satisfying the algebraic condition (∗) [S, V ] = V ^{2}, V e = Se, where S is the multiplication operator f(x) → xf(x), and e is the function with constant value 1. Similarly, if S_{α} is the multiplication operator f → α f on C[0, 1], where α is a given injective real-valued C[0, 1]-function of bounded variation vanishing at 0, then the Stieltjes-Volterra operator f(x) → R ^{x} _{0} f(t) dα(t) on C[0, 1] is characterized as the unique bounded linear operator on the space satisfying the above condition with S = S _{α}. For 1 < p < ∞, the Riemann-Liouville semigroup is characterized as the unique regular semigroup V (·) on ℂ^{+} acting in L^{p}(0, 1), whose boundary group's type is less than π, and for which V := V (1) satisfies Relation (∗).

Original language | English |
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Pages (from-to) | 459-466 |

Number of pages | 8 |

Journal | Acta Scientiarum Mathematicarum |

Volume | 80 |

Issue number | 3-4 |

DOIs | |

State | Published - 2014 |

### Bibliographical note

Publisher Copyright:© Bolyai Institute, University of Szeged.

## Keywords

- Boundary group
- C0-semigroup
- Regular semigroup
- Riemann-Liouville semigroup
- Stieltjes-Volterra operator
- Type (of C0-semigroup)
- Uniqueness theorem
- Volterra Communication Relation
- Volterra operator