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A remark on the frequent hypercyclicity criterion for weighted composition semigroups and an application to the linear von Foerster-Lasota equation. (English) Zbl 1380.47005

The author considers \(C_0\)-semigroups of operators on \(L_p(\Omega,\mu)\), \(1 \leq p < \infty,\) with \(\Omega\) open in \(\mathbb{R}\) and \(\mu\) a Borel measure on \(\Omega\) that admits a strictly positive Lebesgue density. They appear in a natural way when dealing with initial value problems for linear first order partial differential operators. Under mild assumptions, the author extends work by E. M. Mangino and A. Peris [Stud. Math. 202, No. 3, 227–242 (2011; Zbl 1232.47007)] and proves that these \(C_0\)-semigroups satisfy the frequent hypercyclicity criterion if and only if they are chaotic. The result is applied to the linear von Foerster-Lasota \(C_0\)-semigroup on \(L_p(0,1)\) that arises in mathematical biology, to prove that in this case the properties of being hypercyclic, chaotic and frequently hypercyclic are equivalent. In the last section, more general weighted composition \(C_0\)-semigroups on Sobolev spaces are considered.

MSC:

47A16 Cyclic vectors, hypercyclic and chaotic operators
47D06 One-parameter semigroups and linear evolution equations
47F05 General theory of partial differential operators

Citations:

Zbl 1232.47007
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References:

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