Abstract
Let \(\{T(t)\}_{t\ge 0}\) be a continuous semigroup defined on Banach space \(\mathcal {X}\). This paper concerns with continuity and mixing property of \(\{L_{T(t)}\}_{t\ge 0}\), the semigroup which \(\{T(t)\}_{t\ge 0}\) induces on \(K(\mathcal {X})\) (or \(B(\mathcal {X})\)), by left multiplication. We will prove that \(\{L_{T(t)}\}_{t\ge 0}\) is strongly continuous if and only if \(\{T(t)\}_{t\ge 0}\) is too; as well as \(\{L_{T(t)}\}_{t\ge 0}\) has mixing property if and only if \(\{T(t)\}_{t\ge 0}\) has too. It will be seen that weaker results can be obtained when \(\{L_{T(t)}\}_{t\ge 0}\) acts on \(B(\mathcal {X})\).
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Sazegar, A.R., Assadi, A. & Rabieimotlagh, O. Mixing property of \(C_0\)-semigroup of left multiplication operators. Boll Unione Mat Ital 8, 257–263 (2016). https://doi.org/10.1007/s40574-015-0042-0
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DOI: https://doi.org/10.1007/s40574-015-0042-0
Keywords
- Hypercyclic operator
- Left multiplication operator
- Strong operator topology
- Continuous semigroup
- Mixing semigroup