Bulletin of the Belgian Mathematical Society - Simon Stevin

Dynamics of multidimensional Cesáro operators

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

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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


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