Abstract Volterra relation (Q2461614)

A closed operator \(S\) and a bounded operator \(V\) in a complex Banach space \(X\) satisfy the abstract Volterra relation if \[ VD(S) \subseteq D(S) \, , \quad SV - VS \subseteq V^2. \] Various theorems of the second author on semigroups associated with the operators \(V\) and \(S\) are extended here to the case of unbounded Volterra systems \(\{ S_n, V_n: 1 \leq n \leq N\},\) where each pair \((S_n, V_n)\) satisfies the abstract Volterra relation, and to the case where \(V\) is unbounded. Results are on similarity, quasi-affinity and growth of semigroups.