Mixing property of \(C_0\)-semigroup of left multiplication operators. (English) Zbl 1343.47009
The relations between the dynamical properties of a bounded linear operator \(T\) and its corresponding left multiplication operator \(L_T\) were first considered by K. C. Chan and R. D. Taylor [Integral Equations Oper. Theory 41, No. 4, 381–388 (2001; Zbl 0995.46014)] and by J. Bonet et al. [J. Math. Anal. Appl. 297, No. 2, 599–611 (2004; Zbl 1062.47011)].
In the paper under review, the authors consider an analogous problem for the topologically mixing property in the context of \(C_0\)-semigroups, showing that the topologically mixing property is equivalent to be satisfied for a \(C_0\)-semigroup \(\{T_t\}_{t\geq 0}\) and for its corresponding left multiplication \(C_0\)-semigroup \(\{L_{T_t}\}_{t\geq 0}\) in the algebra of compact operators.
In the paper under review, the authors consider an analogous problem for the topologically mixing property in the context of \(C_0\)-semigroups, showing that the topologically mixing property is equivalent to be satisfied for a \(C_0\)-semigroup \(\{T_t\}_{t\geq 0}\) and for its corresponding left multiplication \(C_0\)-semigroup \(\{L_{T_t}\}_{t\geq 0}\) in the algebra of compact operators.
Reviewer: José Alberto Conejero (València)
MSC:
| 47A16 | Cyclic vectors, hypercyclic and chaotic operators |
| 47D06 | One-parameter semigroups and linear evolution equations |
| 47L80 | Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.) |
Keywords:
hypercyclic operator; left multiplication operator; strong operator topology; \(C_0\)-semigroup; topologically mixing semigroupReferences:
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