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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References
Ansari, S.I.: Existence of hypercyclic operators on topological vector spaces. J. Funct. Anal. 148, 384–390 (1997)
Bayart, F., Matheron, E.: Hypercyclic operators failing the hypercyclicity criterion on classical Banach spaces. J. Funct. Anal. 250, 426–441 (2007)
Bonet, J., Martinez-Gimenez, F., Peris, A.: Universal and chaotic multipliers on spaces of operators. J. Math. Anal. Appl. 297, 599–611 (2004)
Chan, K.C., Taylor Jr, R.D.: Hypercyclic subspaces of a Banach space. Integral Equ. Oper. Theory 41, 381–388 (2001)
De la Rosa, M., Read, C.: A hypercyclic operator whose direct sum \(T \oplus T\) is not hypercyclic. J Oper. Theory 61, 369–380 (2009)
Desch, W., Schappacher, W., Webb, G.F.: Hypercyclic and chaotic semigroups of linear operators. Ergod. Theory Dyn. Syst. 17, 793–819 (1997)
Engel, K.J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000)
Fabian, M., Habala, P., Hajek, P., Montesinos, V., Zizler, V.: Banach Space Theory The Basis for Linear and Nonlinear Analysis. Springer, New York (2011)
Godefroy, G., Shapiro, G.: Operators with dense, invariant, cyclic vector manifolds. J. Funct. Anal. 98, 229–269 (1991)
Grosse-Erdman, K.G., Peris, A.: Linear Chaos. Springer, London (2011)
Gupta, M., Mundayadan, A.: Supercyclicity in spaces of operators. Results Math. doi:10.1007/s00025-015-0463-1
Petersson, H.: Hypercyclic conjugate operators. Integral Equ. Oper. Theory 57, 413–423 (2007)
Rolewicz, S.: On orbits of elements. Studia Math. 32, 17–22 (1969)
Zhang, L., Zhou, Z.-H.: Disjointness in supercyclicity on the algebras of Hilbert–Schmidt operators. Indian J. Pure Appl. Math. 46, 219–228 (2015)
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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