tailieunhanh - Frequently hypercyclic weighted backward shifts on spaces of real analytic functions

We study frequent hypercyclicity in the case of weighted backward shift operators acting on locally convex spaces of real analytic functions. We obtain certain conditions on frequent hypercyclicity and linear chaoticity of these operators using dynamical transference principles and the frequent hypercyclicity criterion. | Turk J Math (2018) 42: 3242 – 3249 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Frequently hypercyclic weighted backward shifts on spaces of real analytic functions Berkay ANAHTARCI∗,, Can Deha KARIKSIZ, Department of Natural and Mathematical Sciences, Özyeğin University, İstanbul, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: We study frequent hypercyclicity in the case of weighted backward shift operators acting on locally convex spaces of real analytic functions. We obtain certain conditions on frequent hypercyclicity and linear chaoticity of these operators using dynamical transference principles and the frequent hypercyclicity criterion. Key words: Frequent hypercyclicity, linear chaos, spaces of real analytic functions, weighted backward shifts 1. Introduction Let A(Ω) denote the space of real analytic functions on a nonempty open subset Ω of R , equipped with its natural topology of inductive limit of the spaces H(U ) of holomorphic functions on U , where U runs through all complex open neighborhoods of Ω . Although the spaces H(U ) are Fréchet with the topology of uniform convergence on compact subsets, and monomials form a (Schauder) basis for H(U ), the space A(Ω) is not metrizable and does not admit a Schauder basis [9]. This means that we neither have the Baire category theorem nor a sequential representation for A(Ω) at our disposal, which are important tools in linear dynamics. Due to these obstacles, we define weighted backward shifts on A(Ω) as in [8] by considering their action on monomials using the density of polynomials in A(Ω). For more information on the topology of A(Ω), we refer the reader to the survey of [7]. For the functional analytic tools that we use, we refer the reader to the monograph [12]. Definition A continuous linear operator Bw : A(Ω) → A(Ω) is called a weighted backward shift if Bw