Tuple of operators and hypercyclicity criterion
If \(X\) is a separable Banach space and \((T_{m})_{m\geq 0}\) is a sequence of bounded linear operators on \(X\), the sequence \((T_{m})_{m\geq 0}\) is said to be hypercyclic if there exists a vector \(x\in X\) such that the set \(\{T_{m}x : m\geq 0\}\) is dense in \(X\). When \(T\) is a bounded operator and \(T_{m}=T^{m}\) for each \(m\), then \(T\) itself is said to be hypercyclic. In the present paper as well as in the companion paper [ibid. 76, No.~3, 349--354 (2012; Zbl 1268.47011)], the authors study the case where the sequence \((T_{m})_{m\geq 0}\) is given by all powers \(T_{1}^{k_{1}}T_{2}^{k_{2}}\dots T_{n}^{k_{n}}\), where \(T_{1},\dots, T_{n}\) are \(n\) commuting operators on \(X\), and give versions in this setting of well-known criteria for hypercyclicity.
scientific article; zbMATH DE number 6082512