## Abstract

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 }.

Original language | English |
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Pages (from-to) | 285-292 |

Number of pages | 8 |

Journal | Semigroup Forum |

Volume | 78 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2009 |

## Keywords

- Banach algebra
- C -group of operators
- Lie product
- Perturbation
- Scalar-type
- Similarity
- Similarity orbit
- Spectral operator
- Volterra relation