## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 26, Number 1 (2019), 11-20.

### Dynamics of multidimensional Cesáro operators

J. Alberto Conejero, A. Mundayadan, and J.B. Seoane-Sepúlveda

#### Abstract

We study the dynamics of the multi-dimensional Ces\`aro integral operator on $L^p(I^n)$, for $I$ the unit interval, $1<p<\infty$, and $n\ge 2$, that is defined as \begin{multline*} \displaystyle \mathcal{C}(f)(x_1,\ldots,x_n)=\frac {1} {x_1x_2\cdots x_n} \int_0^{x_1}\ldots\int_{0}^{x_n} f(u_1,\ldots,u_n)du_1\ldots du_n\\ \quad \text{ for } f\in L^p(I^n). \end{multline*} This operator is already known to be bounded. As a consequence of the Eigenvalue Criterion, we show that it is hypercyclic as well. Moreover, we also prove that it is Devaney chaotic and frequently hypercyclic.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 26, Number 1 (2019), 11-20.

**Dates**

First available in Project Euclid: 20 March 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1553047226

**Digital Object Identifier**

doi:10.36045/bbms/1553047226

**Mathematical Reviews number (MathSciNet)**

MR3934078

**Zentralblatt MATH identifier**

07060313

**Subjects**

Primary: 47A16: Cyclic vectors, hypercyclic and chaotic operators 47B37: Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)

Secondary: 47B38: Operators on function spaces (general) 47B99: None of the above, but in this section

**Keywords**

Cesáro integral operator frequent hypercyclicity hypercyclic operator Linear Dynamics

#### Citation

Alberto Conejero, J.; Mundayadan, A.; Seoane-Sepúlveda, J.B. Dynamics of multidimensional Cesáro operators. Bull. Belg. Math. Soc. Simon Stevin 26 (2019), no. 1, 11--20. doi:10.36045/bbms/1553047226. https://projecteuclid.org/euclid.bbms/1553047226