Similarity of operators associated with the Volterra relation

The "Volterra relation" is the commutation relation [S,V] V ², where S is a not necessarily bounded operator, V is a bounded operator leaving D(S) invariant, and [. , .] is the Lie product. When S,V are so related, and in addition iS generates a bounded C ₀-group of operators and V has some general property, it is known that S+α V (α ) is similar to S if and only if α=0 (cf. Theorem 11.17 in Kantorovitz, Spectral Theory of Banach Space Operators, Springer, Berlin, 1983). In particular, S-V is not similar to S. However, it is shown in this note that (without any restriction on V and on the group S() generated by iS), the perturbations (S-V)+P are similar to S for all P in the similarity sub-orbit {S(a)VS(-a);a } of V. When S is bounded, the above perturbations are similar to S for all P in the wider similarity sub-orbit {e ^aS Ve ^-aS ;a }.