Orbits of coanalytic Toeplitz operators and weak hypercyclicity

Abstract

We prove a new criterion of weak hypercyclicity of a bounded linear operator on a Banach space. Applying this criterion, we solve few open questions. Namely, we show that if G is a region of C bounded by a smooth Jordan curve Γ such that G does not meet the unit ball but Γ intersects the unit circle in a non-trivial arc, then M * is a weakly hypercyclic operator on H 2 (G), where M is the multiplication by the argument operator M f (z) = zf (z). We also prove that if g is a non-constant function from the Hardy space H ∞ (D) on the unit disk D such that g(D) ∩ D = ∅ and the set {z ∈ C : |z| = 1, |g(z)| = 1} is a subset of the unit circle T of positive Lebesgue measure, then the coanalytic Toeplitz operator T * g on the Hardy space H 2 (D) is weakly hypercyclic. On the contrary, if g(D) ∩ D = ∅, |g| > 1 almost everywhere on T and log(|g| − 1) ∈ L 1 (T), then T * g is not 1-weakly hypercyclic and hence is not weakly hypercyclic (a bounded linear operator T on a complex Banach space X is called n-weakly hypercyclic if there is x ∈ X such that for every surjective continuous linear operator S : X → C n , the set {S(T m x) : m ∈ N} is dense in C n ). The last result is based upon lower estimates of the norms of the members of orbits of a coanalytic Toeplitz operator. Finally, we show that there is a 1-weakly hypercyclic operator on a Hilbert space, whose square is non-cyclic and prove that a Banach space operator is weakly hypercyclic if and only if it is n-weakly hypercyclic for every n ∈ N.

Hypercyclic Toeplitz Operators

“…However, it seems that hypercyclicity phenomena for general Toeplitz operators are much less studied, and the hypercyclicity criteria are not known. This problem was explicitly stated by Shkarin [10] who described hypercyclic Toeplitz operators with symbols of the form Φ(z) = az + b + cz (i.e., with tridiagonal matrix).…”

“…Univalence of Φ implies that |a| ≥ |c|. To show the strict inequality we need to apply the argument from [10]: if |a| = |c|, then T Φ is a normal operator, and hence is not hypercyclic. In general this argument is not applicable.…”

“…In general this argument is not applicable. On the other hand, in [10] the case |a| < |c| is excluded by appealing to the theory of hyponormal operators. It seems that this kind of argument can not be used for more general operators of the form T γ z+ϕ(z) .…”

“…The property D ∩ (C \ Φ(D)) = ∅ is obvious. To show that D ∩ (C \ Φ(D)) = ∅ one has again to use an ad hoc argument from [10] which uses the very special form of the symbol. Assume, on the contrary, that σ(T Φ ) = C \ Φ(D) ⊂ {z : |z| ≥ 1}.…”

Chaos for the Dynamics of Toeplitz Operators

“…Baranov and Lishanskii [14], inspired by the work of Shkarin [15], studied hypercyclicity of Toeplitz operators on H 2 (D) with symbols of the form p(1/z) + ϕ(z), where p is a polynomial and ϕ ∈ H ∞ . They showed necessary conditions and sufficient conditions for hypercyclicity which almost coincide in the case the degree of p is one.…”