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**Hypercyclic operators; mixing operators; and the bounded steps problem.**
*(English)*
Zbl 1104.47010

This is a very important article on hypercyclic operators \(T\) on separable Banach spaces \(X\) which treats several open problems. First of all, the author solves a problem of Shapiro showing that every infinite-dimensional separable Banach space supports a mixing operator which fails Kitai’s hypercyclicity criterion. An operator \(T\) is called mixing if for every pair \((U,V)\) of non-empty open subsets of \(X\) there is an integer \(N\) such that \(T^n(U)\) meets \(V\) for each \(n \geq N\). Kitai’s criterion can be formulated in an intuitive way by saying that an operator with a dense set of orbits going to 0 and a dense set of backward orbits going to \(0\) is mixing, hence hypercyclic. The assumption of this criterion can be significantly weakened to assure that \(T\) is hypercyclic.

J. Bès and A. Peris [J. Funct.Anal.167, No. 1, 94–112 (1999; Zbl 0941.47002)] proved that \(T\) satisfies (the assumptions of) the weakened hypercyclicity criterion if and only if \(T \oplus T\) is hypercyclic on \(X \oplus X\). Therefore, the hypercyclic criterion problem, i.e., whether every hypercyclic operator \(T \in L(X)\) satisfies the hypercyclicity criterion is equivalent to the following problem of Herrero: is \(T \oplus T\) hypercyclic on \(X \oplus X\) for every hypercyclic operator \(T\in L(X)\)? This problem remains open. However, the author proves that the answer is positive if the operator \(T\) satisfies, in addition, one of the following conditions: (a) \(T\) is upper triangular, (b) \(T\) has a dense set of vectors with bounded orbit, (c) the linear span of the eigenvalues of \(T\) is dense, (d) \(T \oplus T\) is cyclic. The author also proves that if \(T\) is hypercyclic, then \(T \oplus T\) is norm-weak topologically transitive, and she points out that if \(T\) is hypercyclic, then \(T \oplus (-T)\) is cyclic. It is also proved in the paper that an operator \(T\) satisfies the hypercyclicity criterion if and only if for every \(N \geq 2\) (or just for \(N=2\)) and for every sequence of natural numbers \((n_k)_k\) with \(n_{k+1}-n_k \leq N\) for each \(k\), the sequence \((T^{n_k})_k\) is hypercyclic (or universal); a result obtained independently by A. Peris and L. Saldivia in [Integral Equations Oper.Theory 51, No. 2, 275–281 (2005; Zbl 1082.47004)].

The solution of Shapiro’s problem on the existence of mixing operators which fail Kitai’s criterion is based on Salas’ construction of hypercyclic compact perturbations of the identity on \(l_p\), \(1\leq p< \infty\), and a result obtained independently by Atzmon and by Esterle and Zarrabi on orbits with polynomial growth.

There is a misprint in the statement of this result in page 151 of the present paper, an exponent \(1/n\) is missing. The correct result is used in the proofs of Theoems 2.5 and 2.6. However, Peris’ observation Proposition 2.7 seems to depend on the incorrect statement. It is unclear then whether a complex separable hereditarily indecomposable Banach space \(Y\), with hereditarily indecomposable dual, supports an operator which satisfies Kitai’s criterion.

J. Bès and A. Peris [J. Funct.Anal.167, No. 1, 94–112 (1999; Zbl 0941.47002)] proved that \(T\) satisfies (the assumptions of) the weakened hypercyclicity criterion if and only if \(T \oplus T\) is hypercyclic on \(X \oplus X\). Therefore, the hypercyclic criterion problem, i.e., whether every hypercyclic operator \(T \in L(X)\) satisfies the hypercyclicity criterion is equivalent to the following problem of Herrero: is \(T \oplus T\) hypercyclic on \(X \oplus X\) for every hypercyclic operator \(T\in L(X)\)? This problem remains open. However, the author proves that the answer is positive if the operator \(T\) satisfies, in addition, one of the following conditions: (a) \(T\) is upper triangular, (b) \(T\) has a dense set of vectors with bounded orbit, (c) the linear span of the eigenvalues of \(T\) is dense, (d) \(T \oplus T\) is cyclic. The author also proves that if \(T\) is hypercyclic, then \(T \oplus T\) is norm-weak topologically transitive, and she points out that if \(T\) is hypercyclic, then \(T \oplus (-T)\) is cyclic. It is also proved in the paper that an operator \(T\) satisfies the hypercyclicity criterion if and only if for every \(N \geq 2\) (or just for \(N=2\)) and for every sequence of natural numbers \((n_k)_k\) with \(n_{k+1}-n_k \leq N\) for each \(k\), the sequence \((T^{n_k})_k\) is hypercyclic (or universal); a result obtained independently by A. Peris and L. Saldivia in [Integral Equations Oper.Theory 51, No. 2, 275–281 (2005; Zbl 1082.47004)].

The solution of Shapiro’s problem on the existence of mixing operators which fail Kitai’s criterion is based on Salas’ construction of hypercyclic compact perturbations of the identity on \(l_p\), \(1\leq p< \infty\), and a result obtained independently by Atzmon and by Esterle and Zarrabi on orbits with polynomial growth.

There is a misprint in the statement of this result in page 151 of the present paper, an exponent \(1/n\) is missing. The correct result is used in the proofs of Theoems 2.5 and 2.6. However, Peris’ observation Proposition 2.7 seems to depend on the incorrect statement. It is unclear then whether a complex separable hereditarily indecomposable Banach space \(Y\), with hereditarily indecomposable dual, supports an operator which satisfies Kitai’s criterion.

Reviewer: José Bonet (Valencia)

### MSC:

47A16 | Cyclic vectors, hypercyclic and chaotic operators |