On hypercyclic operators in weighted spaces of entire functions

A. I. Rakhimova

Taurida Journal of Computer Science Theory and Mathematics, no. 1 (2023), pp. 88-110

Résumé

In this paper we have studied various hypercyclic operators in the weighted space of entire functions $\mathcal{F}_{\varphi}$. Hypercyclic operators play an important role in the theory of dynamical systems. Note that questions about hypercyclic operators have been considered in detail only in spaces of entire and analytic functions. And in the weighted spaces of entire functions, such operators are not yet very well studied and provide a large number of research problems. The space $\mathcal{F}_{\varphi}$ is defined as follows. Let $\varphi=\{\varphi_m(z)\}_{m=1}^\infty$ — arbitrary family of functions convex in $\mathbb{C}^n$ taking real values and satisfying some conditions on their growth $i_1)$–$i_4)$. Now for each function we introduce a weighted normed space $\mathcal{F}_{m}$ consisting of from integer functions in $\mathbb{C}^n$. Let $\mathcal{F}_{\varphi}$ denote their projective limit. Then it is a Frechet–Schwarz space of type $(FS)$. Next for the space $\mathcal{F}_{\varphi}$ one can find additional conditions on the weight functions, under which it will be invariant under differentiation. It can also be shown that, under the same conditions, it is shift-invariant. Then we can consider the problems of hypercyclicity in it partial differentiation and shift operators, their compositions, convolution operator and operators commuting with differentiation. Theorem 1 proves hypercyclicity in the space $\mathcal{F}_{\varphi}$ partial differentiation operator $\frac{\partial}{\partial z_j}, \quad j\in \{1;n\}$ with respect to any of the complex variables. In Theorem 2 hypercyclicity in this space of a finite sum of such operators was shown $\sum\limits_{\alpha\in\mathbb{Z}_+^n: \,\, |\alpha| \leq m}c_\alpha D_z^\alpha f$, and in Theorem 3 — their infinite sum $\sum\limits_{\alpha\in\mathbb{Z}_+^n: \,\, |\alpha| \geq 0}c_\alpha D_z^\alpha f$ under certain conditions on the coefficients of the series. Theorem 4 states that the shift operator by some constant $T_a:\,f(z)\in\mathcal{F}_\varphi \longrightarrow f(z+a)$, where $a\in\mathbb{C}^n$, and it is not equal to zero, is hypercyclic in $\mathcal{F}_{\varphi}$. By Theorem 5 it follows that a continuous linear operator $T$ in the space ${\mathcal F}_\varphi$, which commutes with partial differentiation operators and is not equal to a scalar multiple of the identity mapping, is hypercyclic in the given space. The following corollaries follow from this theorem. The finite sum of shifts and the sum of compositions of shifts with partial differentiation operators are hypercyclic. Also, under certain requirements for the coefficients of the series, the infinite sum of shifts will be hypercyclic. Theorem 6 considers an operator of the form $Tf(z)=\sum\limits_{j=1}^n c_j \frac{\partial}{\partial z_j}(f(\lambda z+b))$, where all numbers $\lambda\in\mathbb{C}$, $b\in\mathbb{C}^n$ and $c_j\in\mathbb{C}^n,\,\,j\in\{1 ;n\}$ fixed. Then this operator is hypercyclic in the space $\mathcal{F}_\varphi$ under the condition $|\lambda| \leq 1$. Next in Theorem 7 consider a generalized function with compact support $S$, whose Fourier-Laplace transform is not identical to a constant value. Then we can make sure that the convolution operator of the form $M_S[f](z) = S_t(f(z+t))$ is hypercyclic in $\mathcal{F}_\varphi$. We also consider the case when the additional condition are satisfied for the family of functions $\varphi$, and $S$ is defined as a continuous linear functional on $\mathcal{F}\varphi$. Then the convolution operator $M_S[f]$ will also be hypercyclic in $\mathcal{F}\varphi$.

Détail

Keywords: hypercyclic operator, weight space, partial differentiation operator, shift operator, convolution operator, entire function.