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Existence of Linear Hypercyclic Operators on Infinite-Dimensional Banach Spaces

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Existence of Linear Hypercyclic Operators on Infinite-Dimensional Banach Spaces Existence of Linear Hypercyclic Operators on Infinite-Dimensional Banach Spaces Kuikui Liu June 8, 2015 Contents 1 Introduction 2 2 Preliminaries 2 2.1 Metric Spaces and Completeness . 2 2.2 Linear Spaces and Norms . 2 2.3 Separability . 3 2.4 Banach Spaces, Linear Operators and Dual Spaces . 3 2.5 Orbits and Hypercyclicity . 5 3 The Existence Theorem and its Proof 5 3.1 Intermediate Results . 5 3.2 The Existence Theorem . 7 4 Hypercyclicity in a Different Context 11 4.1 Hypercyclicity in F-Spaces . 11 4.2 Open Problems . 12 5 Dynamics and Chaos 12 5.1 Dynamics . 12 5.2 Chaos . 13 6 Conclusion 14 1 1 Introduction The notion of hypercyclicity was first introduced by operator theorists studying the connections between cyclic vectors and the invariant subspace problem, which, for Hilbert spaces (Banach spaces where the norm is induced from an inner product), remains an open problem (Enflo constructed a Banach space and a linear operator that does not a have nontrivial invariant subspace [7]). The invariant subspace problem asks if every bounded linear operator on a space possesses a nontrivial, closed invariant subspace. A vector is cyclic with respect to a bounded linear operator if the span of its orbit is dense in the containing space. Hypercyclic vectors are a special case of cyclic vectors and are related to a similar problem called the invariant subset problem, although, this isn't discussed in this paper In 1969, S. Rolewicz posed the problem \Does an infinite-dimensional, separable Banach space support a hypercyclic operator?" [6]. It was proven later that the answer is the affirmative. In this paper, we examine and summarize a proof of this existence result, which was given by L. Bernal-Gonz´alez[1]. We begin by setting preliminary definitions and terminology [1][4]. Then, we provide some related intermediate results and then present the existence theorem. A proof is given using those intermediate results. Afterwards, we present some results on hypercyclicity in a related context along with open problems and discuss applications of hypercyclicity to dynamics and chaos. 2 Preliminaries We begin this paper by laying out the necessary preliminary definitions that will be needed. 2.1 Metric Spaces and Completeness Definition 2.1. Let X be a nonempty set. A metric (or distance function) is a function d : X × X ! R that satisfies the following properties: • d(x; y) = 0 if and only if x = y • d(x; y) ≥ 0 for all x; y 2 X (non-negativity) • d(x; y) = d(y; x) for all x; y 2 X (symmetry) • d(x; z) ≤ d(x; y) + d(y; z) for all x; y; z 2 X (Triangle Inequality) The pair (X; d) is called a metric space. When there is no ambiguity, X will be used instead to denote a metric space. Definition 2.2. Let X be a metric space. A sequence of points fxng in X converges to a point x 2 X if for every > 0, there exists N 2 N such that d(xn; x) < for all n > N. Definition 2.3. Let X be a metric space. A sequence of points fxng in X is Cauchy if for any > 0, there exists N 2 N such that d(xm; xn) < for all m; n 2 N. Remark. Convergent sequences are Cauchy. Choose N 2 N large such that d(xn; x) < /2 when n > N. Then, d(xm; xn) ≤ d(xm; x) + d(x; xn) < /2 + /2 = for all m; n > N. Definition 2.4. A metric space is said to be complete if every Cauchy sequence of points fxng in X converges to a point x 2 X. 2.2 Linear Spaces and Norms Definition 2.5. A linear space (or vector space) V over a scalar field K (for our purposes, K = R or K = C) is a set of points (or vectors) on which element addition and scalar multiplication are defined. They must satisfy the addition axioms given above as well as the following. 1. If x; y 2 V and λ, µ 2 K, then λx + µy 2 V (closure under vector addition and scalar multiplication) 2 2. If x; y 2 V and λ, µ 2 K, then λ(µx) = (λµ)x, λ(x + y) = λx + λy and (λ + µ)x = λx + µx. Definition 2.6. If V is a linear space, then a seminorm on V is a function f : V ! R with following properties: • f(x) ≥ 0 for all x 2 V (non-negativity) • If x 2 V and λ 2 K, then f(λx) = jλjf(x) • f(x + y) ≤ f(x) + f(y) for all x; y 2 V (Triangle Inequality) If, in addition, f(x) = 0 if and only if x = 0, then f is called a norm and is more commonly denoted by jj · jj : V ! R. Intuitively, norms define a notion of vector \length" or \magnitude". In cases of possible confusion, we write jjxjjV instead of jjxjj to denote the norm of x 2 V with respect to the norm defined on V . Definition 2.7. A normed linear space is a pair (V; jj · jj), where V is a linear space and jj · jj is a norm. Normed linear spaces are metric spaces with respect to the metric d(x; y) = jjx − yjj. Definition 2.8. Let X be a linear space and S = fv1; : : : ; vng ⊂ X. A vector w is a linear combination of the vectors v1; : : : ; vn if there exist scalars c1; : : : ; cn 2 K such that n X w = ckvk k=1 Definition 2.9. Let X be a linear space and S ⊂ X. We define the span of S to be span S = fw : w = c1v1 + ··· + ckvk for v1; : : : ; vk 2 S and c1; : : : ; ck 2 Kg Note that in the above definition, the linear combinations are taken over a finite set of vectors, even if S is infinite. 2.3 Separability Definition 2.10. Let X be a metric space and E be a subset of X. A point x 2 X is a limit point of E if there exists a sequence of points fxng in E that converge to x. Definition 2.11. Let E be a subset of a metric space X. E is dense in X if every point of X is a limit point of E. Equivalently, E is dense in X if E = X since E is the union of E and all of its limit points. Definition 2.12. A metric space X is separable if there exists a dense subset E ⊂ X that is countable. 2.4 Banach Spaces, Linear Operators and Dual Spaces Definition 2.13. A Banach space is a complete, normed linear space. Definition 2.14. Let X and Y be Banach spaces over a scalar field K. T : X ! Y is a linear operator if • T (x + y) = T (x) + T (y) for all x; y 2 X • T (λx) = λT (x) for all x 2 X and λ 2 K We often write T x instead of T (x). It is clear that compositions of linear operators are linear. Indeed, if S : X ! Y and T : Y ! Z are two linear operators, where X, Y and Z are Banach spaces, then (S ◦ T )(x + y) = S(T (x + y)) = S(T (x) + T (y)) = S(T (x)) + S(T (y)) = (S ◦ T )(x) + (S ◦ T )(y) and (S ◦ T )(λx) = S(T (λx)) = S(λT (x)) = λS(T (x)) = λ(S ◦ T )(x) so that the composition S ◦ T of S and T is also a linear operator. 3 Definition 2.15. Let X and Y be two Banach spaces. A linear operator T : X ! Y is bounded if there exists positive C 2 R such that jjT xjj ≤ Cjjxjj for all x 2 X. Otherwise, T is unbounded. We denote the set of linear operators by L(X; Y ) and the set of bounded linear operators by B(X; Y ). If X = Y , we simply write L(X) and B(X), respectively. In general, it isn't obvious that all linear operators are bounded. For finite-dimensional spaces, this is the case. However, there are many examples of unbounded linear operators in infinite-dimensional spaces; the boundedness also depends on the norms used. As an example, let C1([0; 1]) be the set of continuously differentiable functions [0; 1] and C0([0; 1]) be the set of continuous functions on [0; 1]. If both are equipped with the sup-norm jjfjj1 = supx2[0;1] jf(x)j, then the differentiation operator D : C1([0; 1]) ! C0([0; 1]) is unbounded. To see this, note that the set of n 1 pn(x) = x (n ≥ 1) is a subset of C [0; 1]. Furthermore, jjpnjj = 1 for all n. However, jjDpnjj1 = jjnpn−1jj1 = njjpn−1jj1 = n which is unbounded. If instead, we equip C1([0; 1]) with the C1 norm, given by 0 jjfjjC1 = sup jf(x)j + sup jf (x)j x2[0;1] x2[0;1] (and C0([0; 1]) with the sup-norm), then D : C1([0; 1]) ! C0([0; 1]) is bounded. To see this, 0 0 jjDfjj1 = sup jf (x)j ≤ sup jf(x)j + sup jf (x)j = jjfjjC1 x2[0;1] x2[0;1] x2[0;1] So, jjDjj = 1 in this case. Definition 2.16. If T : X ! Y is a bounded linear operator, then we define the operator norm (or uniform norm) jjT jj of T by jjT jj = inffM : 8x 2 X; jjT xjj ≤ Mjjxjjg The operator norm defines a norm on B(X; Y ), which can be verified as follows.
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