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**The hypercyclicity criterion for sequences of operators.**
*(English)*
Zbl 1032.47006

Let \(X\) denote a separable, complete, metrizable topological vector space (a separable \(F\)-space). A sequence \((T_n) \subset L(X)\) of continuous linear operators on \(X\) is called hypercyclic if there exists \(x \in X\), called a hypercyclic vector, for the sequence, such that its orbit \(\{ T_1(x),T_2(x),\dots \}\) is dense in \(X\). A sequence \((T_n)\) is said to satisfy the hypercyclicity criterion if there are dense subsets \(X_0\) and \(Y_0\) of \(X\) and an increasing sequence \((n(k))_k\) of natural numbers such that (i) \(T_{n(k)}(x)\) tends to \(0\) for each \(x \in X_0\), and (ii) for each \(y \in Y_0\) there is a sequence \((u_k)\) converging to \(0\) in \(X\) such that \(T_{n(k)}(u_k)\) tends to \(y\). According to the authors, the “great open problem” of hypercyclicity asks whether the hypercyclicity criterion holds for \((T^n)\) if \(T\) is an operator such that the sequence \((T^n)\) of its iterates is hypercyclic.

This interesting paper presents extensions of deep results due to J. Bès and A. Peris [J. Funct. Anal. 167, 94-112 (Zbl 0941.47002)]. The sequence of operators \((T_n)\) satisfies the hypercyclicity criterion if and only if \((T_n)\) is densely hereditarily hypercyclic (see the definition in the article), and if and only if, for each \(N\), the sum \((T_n \oplus \dots \oplus T_n)\), \(N\)-fold, has a dense set of hypercyclic vectors on \(X^N\). Applications of this result are given to hypercyclic sequences of composition operators on the space \(H(G)\) of analytic functions, thus extending previous work by Bernal and Montes.

Further characterizations are obtained if the sequence \((T_n)\) is assumed to be almost commuting. In particular, it is proved that the sequence of iterates \((T^n)\) of an operator \(T\) satisfies the hypercyclicity criterion if and only if for each pair \(U,V\) of non-empty open subsets of \(X\) and every neighbourhood \(W\) there is \(n\) such that both \(T^n(U)\) and \(T^n(V)\) meet \(W\). A chaotic weakly commuting sequence of operators satisfies the hypercyclicity criterion.

This interesting paper presents extensions of deep results due to J. Bès and A. Peris [J. Funct. Anal. 167, 94-112 (Zbl 0941.47002)]. The sequence of operators \((T_n)\) satisfies the hypercyclicity criterion if and only if \((T_n)\) is densely hereditarily hypercyclic (see the definition in the article), and if and only if, for each \(N\), the sum \((T_n \oplus \dots \oplus T_n)\), \(N\)-fold, has a dense set of hypercyclic vectors on \(X^N\). Applications of this result are given to hypercyclic sequences of composition operators on the space \(H(G)\) of analytic functions, thus extending previous work by Bernal and Montes.

Further characterizations are obtained if the sequence \((T_n)\) is assumed to be almost commuting. In particular, it is proved that the sequence of iterates \((T^n)\) of an operator \(T\) satisfies the hypercyclicity criterion if and only if for each pair \(U,V\) of non-empty open subsets of \(X\) and every neighbourhood \(W\) there is \(n\) such that both \(T^n(U)\) and \(T^n(V)\) meet \(W\). A chaotic weakly commuting sequence of operators satisfies the hypercyclicity criterion.

Reviewer: José Bonet (Valencia)

### MSC:

47A16 | Cyclic vectors, hypercyclic and chaotic operators |

46A16 | Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.) |

46A04 | Locally convex Fréchet spaces and (DF)-spaces |