When is a scaled contraction hypercyclic?
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When is a scaled contraction hypercyclic? Valentin Matache1 Received: 6 March 2020 / Accepted: 12 September 2020 © Unione Matematica Italiana 2020
Abstract Hypercyclic operators are operators with dense orbits. A contraction cannot be hypercyclic since its orbits are bounded sets. Nevertheless, by multiplying a contraction with a scalar of absolute value larger than 1, the resulting scaled contraction can occasionally be a hypercyclic operator. In this paper, we investigate which Hilbert space contractions have that property and which don’t. We introduce the set (T ) of all scalars which produce a hypercyclic operator, by scaling the operator T , and determine (T ) in various cases. New properties of hyperciclic operators are discovered in this process. For instance, it is proved that any connected component of the essential spectrum of a hypercyclic operator must meet the unit circle. Keywords Hypercyclic operators · Contractions · Shifts Mathematics Subject Classification Primary 47A16 · Secondary 47A20
1 Introduction This is a paper inspired by an example of Rolewicz (Example 1, below). While a contraction, that is an operator T on a normed space, with property T ≤ 1, cannot have dense orbits, for the mere reason that the orbits of such an operator are bounded sets, scaling the aforementioned operator by scalars of absolute value larger than 1 can produce hypercyclic operators, that is operators with dense orbits. The question is: which contractions have that property? Surely, not all of them. Here are some details. Let X denote a Hausdorff, complex, linear topological space, and L(X ) the algebra of linear, continuous operators acting on X . For all x ∈ X , the set
OT (x) := {x, T x, T 2 x, . . . , T n x, . . . }
Dedicated to the memory of Ciprian Foia¸s
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Valentin Matache [email protected] Department of Mathematics, University of Nebraska, Omaha, NE 68182, USA
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V. Matache
is called the orbit of x under T . If OT (x) is dense, then x is called a hypercyclic vector of T . Operators having hypercyclic vectors are called hypercyclic operators. Clearly, such operators can exist only on separable spaces. Also, it has been proved in [13] (see also [15]), that: Theorem 1 Nonzero, finite dimensional, Hausdorff, linear topological spaces cannot support hypercyclic operators. For that reason, the hypercyclicity of operators will be studied in the sequel only on complex, infinite–dimensional, separable spaces. For the basic facts on hypercyclic operators, the reader is referred to [7]. Sometimes, in this paper, the reader is referred to the papers where results on such operators were originally obtained. Most of those results can also be found (with proofs) in the book [7], a fact we emphasize for the benefit of the reader. Recall that a Hilbert space isometry V ∈ L(H ) is called a unilateral forward shift if V ∗n → 0 strongly in H . In that case, V ∗ , the adjoint of V , is called a unilateral backward shift. The Hilbert dimension of the kernel of V ∗ is called the multiplicity of V or of V ∗ . The w
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