A hypercyclic operator whose adjoint is also hypercyclic

TL;DR: In this paper, the existence of a bilateral weighted shift whose adjoint is also hypercyclic was shown, based on the similarity orbit theorem, which is a key result of Herrero's work.

Abstract: An operator T acting on a Hubert space H is hypercyclic if, for some vector x in H, the orbit {Tnx: n > 0} is dense in H. We show the existence of a hypercyclic operator—in fact, a bilateral weighted shift—whose adjoint is also hypercyclic. This answers positively a question of D. A. Herrero. Introduction Let H be a complex, separable, infinite-dimensional Hubert space. Let B(H) denote the bounded linear operators acting on H. Let T e B(H) and x e H. The orbit of x under T is Orb(7\ x) — {x, Tx ,T x, ...} . T is said to be cyclic if there exists y e H such that the linear span of Oxb(T, y) is dense in H ; T is hypercyclic if Orb(T, y) itself is dense in H. In this last case, y is hypercyclic for T (in [2] such a y is called universal, and in [4] such a y is called orbital). In a recent paper [7], D. A. Herrero completely characterized the closure of the class of hypercyclic operators in B(H) in terms of spectral properties. His proof is based on previous work of C. Kitai [10], Godefroy and Shapiro [3], and Herrero [5] and [6]. Also a key result that he uses is the similarity orbit theorem [1]. In [7], Herrero raises several questions. The purpose of this note is to answer positively Question 2: "Does there exist T e B(H) such that both T and T* are hypercyclic?" We also answer in the affirmative Question 3: "Does there exist a hypercyclic backward shift such that bQbx ■ ■ ■ bk_x does not tend to infinity? (Here the b's are the weights of the shift.)" Gethner and Shapiro proved in [2] that if the weights bk > 1 and b0--bk go to infinity, then the backward shift is hypercyclic. In [7], it is shown that the condition bk > 1 can be dropped. Observe that if U is the unilateral (unweighted) shift, 2U* is hypercyclic, while 2I7 is not. Received by the editors January 2, 1990 and, in revised form, April 18, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 47B99; Secondary 47B37.