## 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.

### MSC:

 47A16 Cyclic vectors, hypercyclic and chaotic operators

### Citations:

Zbl 0941.47002; Zbl 1082.47004