For a topological vector space x, lx is the algebra of continuous linear operators on x, x. Let be a topological vector space, and let be the algebra of continuous linear operators on. Alexandre publication date 1973 topics linear topological spaces. In particular, it answers in the affirmative a question of s.
In mathematics, especially functional analysis, a hypercyclic operator on a banach space x is a bounded linear operator t. Hypercyclic and topologically transitive semigroups of. Important concepts in linear dynamics are that of a hypercyclic operator. The operators are disjoint hypercyclic if there is such that the orbit is dense in. Existence and nonexistence of hypercyclic semigroups.
By an operator, we always mean a continuous linear operator. Rolewicz, posed in 1969, whether or not every infinite. For example, the derivation operator and the nontrivial translation operators on. Hypercyclic tuples of operators were introduced in 5, 7 and 12. X such that the the orbit of xunder the action of the semigroup generated by t1. The direct sum of two hypercyclic operators is not in general a hypercyclic operator. F space that is, a complete and metrisable topological vector space over the scalar field k i or c and denote by le the space of all continuous linear operators on e. A corollary of theorem 1c is that all complete metrizable locally convex spaces in particular all banach spaces admit continuous hypercyclic operators. In other words, the smallest closed invariant subset containing x is the whole space. In this last paper, these operators have allowed to exhibit, among others, frequently hypercyclic operators which are not ergodic or ufrequently hypercyclic operators which are not frequently hypercyclic on hilbert spaces. Ansaris proof that every operator on a complex banach space shares with its powers the same hypercyclic vectors 2, thm 1 works for the real scalar case as well see also 1, note 3. However, each of these counterexamples were not invertible and it will be necessary to adapt several results relating on. Hypercyclic operators on countably dimensional spaces schenke, a.
An operator t le is said to be cyclic if there is a vector x e, called cyclic vector. Introduction a continuous linear operator ton topological vector space xis said to be hypercyclic provided there is an x2xwhose orbit under t. Frequently hypercyclic weighted backward shifts on spaces of real analytic functions. Lx is said to be hypercyclic if there exists some x. Hence, every topological vector space is an abelian topological group. Assume that x is a topological vector space over the field k jr or c. One can adopt a purely topological viewpoint, investigating in particular the individual behaviour of orbits. Birkhoff proved the existence of a hypercyclic operator on a certain complete metrizable locally convex space. Thypercyclic, and the set of all hypercyclic vectors for t is denoted by hct. We recall that a continuous linear operator t on a topological vector space x is hypercyclic if there is a vector x in x such that the set ftnx.
Hypercyclic operators on topological vector spaces. Pdf hypercyclic operators on topological vector spaces. Common hypercyclic functions for translation operators with large gaps ii. We prove that, for a wide class of topological vector spaces, every sotdense set of operators is hypertransitive. Such a vector is called a hypercyclic vector for and the set of hypercyclic vectors for will be denoted by.
By continuing to use our website, you are agreeing to our use of cookies. A basic notion in this context is that of hypercyclicity. Existence of hypercyclic operators on topological vector. Topological vector spaces the continuity of the binary operation of vector addition at 0,0 in v. The earliest examples of hypercyclic operators were operators on the space hc of entire functions. We say that t is hypercyclic if, for some x in e, the orbit of x on t, orbx,tx, tx, t2 x. The invariant subspace problem asks if every bounded linear operator on a space possesses a nontrivial, closed invariant subspace. Finally, condition is necessary for the existence of afrequently hypercyclic operators on banach spaces. Let x be a complex topological vector space with dimx 1 and bx the set of all continuous linear operators on x. Common hypercyclic functions for translation operators. In particular, if xis a topological vector space with a countable open basis u n n 1, we deduce that in order to prove that a vector xis afrequently hypercyclic, it is su cient to prove that nx.
Introduction let x be a topological vector space over kr or c. Feldman, who have raised 7 questions on hypercyclic tuples. In general, however, the reader will lose very little on assuming that we are working in banach spaces. Mixing operators on spaces with weak topology queens. Grosseerdmann, introduction to linear dynamics a linear dynamical system is given by a continuous linear operator ton a topological vector space x. V is equivalent to the statement that for each open subset u1 of v such that 0. A continuous linear operator acting on a topological vector space is called hypercyclic, if there exists a vector such that the orbit of under is dense in. We show that if x2x has orbit under t that is somewhere dense in x, then the orbit of xunder t must be everywhere dense in x, answering a question raised by alfredo peris. The present paper introduces a very simple, but very useful notion of the so called quasiextension of l1 operators and proves that a large class of topological vector spaces admit continuous hypercyclic operators.
Recall that a topological vector space is a vector space together with a. Hypercyclic operators and rotated orbits with polynomial. Inverse of ufrequently hypercyclic operators sciencedirect. An operator t on x is said to be hypercyclic provided there exists a vector x in x such that the orbit orbx tnx. University of crete, department of mathematics and applied mathematics, crete, greece. If a topological space supports a hypercyclic transformation then it is necessarily separable. We give a partial description for non necessarily finite dimensional subsets.
Frequently hypercyclic weighted backward shifts on spaces. Existence of linear hypercyclic operators on in nite. Bonet, frerick, peris and wengenroth constructed a hypercyclic operator on the locally convex direct sum of countably many copies of the banach space we use cookies to enhance your experience on our website. Let lx denote the space of all operators on x, that is, all continuous linear mappings x x. The operator t is said to be hypercyclic if there is some vector x. We give a few observations on different types of bounded operators on a topological vector space x and their relations with compact operators on x. Throughout the article, all topological spaces are assumed to be hausdorff. Introduction all vector spaces in this article are assumed to be over k being either the. Hypercyclic operators on topological vector spaces core. Our aim will be to study some results about hypercyclicity and to observe how some spaces behave regarding this class of operators. We provide extensions of this result for orbits of operators which are rotated by unimodular complex numbers with polynomial phases.
N2 we prove that a continuous linear operator t on a topological vector space x with weak topology is mixing if and only if the dual operator t has no finite dimensional invariant subspaces. A tuple t1, tn of commuting continuous linear operators on a topological vector space x is called hypercyclic if there is x. On supercyclicity of operators from a supercyclic semigroup. Let x be a separable f space namely a topological vector space whose topology is induced by a complete invariant metric. Throughout this paper all topological spaces and topological vector spaces areassumedtobehausdorff. Hypercyclic operators failing the hypercyclicity criterion. A vector is cyclic with respect to a bounded linear operator if the span of its orbit is dense in the containing space. X such that the orbit of x under the action of the semigroup.
We study hypercyclicity, devaney chaos, topological mixing properties and strong mixing in the measuretheoretic sense for operators on topological vector spaces with invariant sets. Weakly mixing operators on topological vector spaces. Hypercyclic tuples of the adjoint of the weighted composition operators rahmat soltani, bahram khani robati, karim hedayatian. In these notes, the main object of study is linear hypercyclic transformations on a topological vector space. Ansari,existence of hypercyclic operators on topological vector spaces,j. Hypercyclic tuples of operators on cn and rn stanislav shkarin abstract a tuple t1. We answer those 4 of them, which can be dealt with on the level of operators on finite dimensional spaces. In particular, we show that if e is a normed vector space and x x is onetoone and runaway on x, then the composition operator f. Hypercyclic operators on countably dimensional spaces. Then an operator t e lx is called hypercyclic whenever there exists some x c x such that the orbit tnx.