University of crete, department of mathematics and applied mathematics, crete, greece. Rolewicz, posed in 1969, whether or not every infinite. Hypercyclic operators and rotated orbits with polynomial. The operators are disjoint hypercyclic if there is such that the orbit is dense in.
However, each of these counterexamples were not invertible and it will be necessary to adapt several results relating on. Hypercyclic and topologically transitive semigroups of. On supercyclicity of operators from a supercyclic semigroup. In particular, it answers in the affirmative a question of s. Grosseerdmann, introduction to linear dynamics a linear dynamical system is given by a continuous linear operator ton a topological vector space x. Lx is said to be hypercyclic if there exists some x. 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. We prove that, for a wide class of topological vector spaces, every sotdense set of operators is hypertransitive. The earliest examples of hypercyclic operators were operators on the space hc of entire functions. Hence, every topological vector space is an abelian topological group. Introduction let x be a topological vector space over kr or c. Weakly mixing operators on topological vector spaces. In other words, the smallest closed invariant subset containing x is the whole space.
The operator t is said to be hypercyclic if there is some vector x. Introduction a continuous linear operator ton topological vector space xis said to be hypercyclic provided there is an x2xwhose orbit under t. We study hypercyclicity, devaney chaos, topological mixing properties and strong mixing in the measuretheoretic sense for operators on topological vector spaces with invariant sets. Hypercyclic tuples of operators were introduced in 5, 7 and 12. Throughout the article, all topological spaces are assumed to be hausdorff. Topological vector spaces the continuity of the binary operation of vector addition at 0,0 in v. The invariant subspace problem asks if every bounded linear operator on a space possesses a nontrivial, closed invariant subspace. 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. Finally, condition is necessary for the existence of afrequently hypercyclic operators on banach spaces. Frequently hypercyclic weighted backward shifts on spaces. Then an operator t e lx is called hypercyclic whenever there exists some x c x such that the orbit tnx. We give a partial description for non necessarily finite dimensional subsets.
In these notes, the main object of study is linear hypercyclic transformations on a topological vector space. 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. Existence of linear hypercyclic operators on in nite. One can adopt a purely topological viewpoint, investigating in particular the individual behaviour of orbits. A vector is cyclic with respect to a bounded linear operator if the span of its orbit is dense in the containing space. A corollary of theorem 1c is that all complete metrizable locally convex spaces in particular all banach spaces admit continuous hypercyclic operators. 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. X such that the the orbit of xunder the action of the semigroup generated by t1. 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. An operator t on x is said to be hypercyclic provided there exists a vector x in x such that the orbit orbx tnx. Important concepts in linear dynamics are that of a hypercyclic operator. A vector space is an abelian group with respect to the operation of addition, and in a topological vector space the inverse operation is always continuous since it is the same as multiplication by. Let x be a separable f space namely a topological vector space whose topology is induced by a complete invariant metric. Hypercyclic tuples of the adjoint of the weighted composition operators rahmat soltani, bahram khani robati, karim hedayatian.
We answer those 4 of them, which can be dealt with on the level of operators on finite dimensional spaces. Let be a topological vector space, and let be the algebra of continuous linear operators on. 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. If a topological space supports a hypercyclic transformation then it is necessarily separable. X such that the orbit of x under the action of the semigroup. Existence and nonexistence of hypercyclic semigroups. We say that t is hypercyclic if, for some x in e, the orbit of x on t, orbx,tx, tx, t2 x. Hypercyclic tuples of operators on cn and rn stanislav shkarin abstract a tuple t1. Throughout this paper all topological spaces and topological vector spaces areassumedtobehausdorff.
Common hypercyclic functions for translation operators. Inverse of ufrequently hypercyclic operators sciencedirect. Feldman, who have raised 7 questions on hypercyclic tuples. Birkhoff proved the existence of a hypercyclic operator on a certain complete metrizable locally convex space. Throughout, by an operator we mean a continuous linear mapping. Mixing operators on spaces with weak topology queens. The direct sum of two hypercyclic operators is not in general a hypercyclic operator. Let lx denote the space of all operators on x, that is, all continuous linear mappings x x. 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. Hypercyclic operators on topological vector spaces core. For example, the derivation operator and the nontrivial translation operators on. Rate of growth of frequently hypercyclic functions. 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.
Abstract let hc be the set of entire functions endowed with the topology of local uniform convergence. Hypercyclic operators on countably dimensional spaces schenke, a. A tuple t1, tn of commuting continuous linear operators on a topological vector space x is called hypercyclic if there is x. Such a vector is called a hypercyclic vector for and the set of hypercyclic vectors for will be denoted by. Our aim will be to study some results about hypercyclicity and to observe how some spaces behave regarding this class of operators. Thypercyclic, and the set of all hypercyclic vectors for t is denoted by hct. Assume that x is a topological vector space over the field k jr or c. Introduction all vector spaces in this article are assumed to be over k being either the. By continuing to use our website, you are agreeing to our use of cookies. 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. Existence of hypercyclic operators on topological vector. 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. Hypercyclic operators on countably dimensional spaces.
An operator t le is said to be cyclic if there is a vector x e, called cyclic vector. Pdf hypercyclic operators on topological vector 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. V is equivalent to the statement that for each open subset u1 of v such that 0. In mathematics, especially functional analysis, a hypercyclic operator on a banach space x is a bounded linear operator t. Let x be a complex topological vector space with dimx 1 and bx the set of all continuous linear operators on x. Ansari,existence of hypercyclic operators on topological vector spaces,j. By an operator, we always mean a continuous linear operator. Hypercyclic operators on topological vector spaces. For a topological vector space x, lx is the algebra of continuous linear operators on x, x. In general, however, the reader will lose very little on assuming that we are working in banach spaces. We provide extensions of this result for orbits of operators which are rotated by unimodular complex numbers with polynomial phases.
Alexandre publication date 1973 topics linear topological spaces. Common hypercyclic functions for translation operators with large gaps ii. A basic notion in this context is that of hypercyclicity. Hypercyclic operators failing the hypercyclicity criterion. Recall that a topological vector space is a vector space together with a.
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