Abstract
Grosse-Erdmann and Kim recently introduced the notion of bihypercyclicity for studying the existence of dense orbits under bilinear operators. We propose an alternative notion of orbit for -linear operators that is inspired by difference equations. Under this new notion, every separable infinite dimensional Fréchet space supports supercyclic -linear operators, for each . Indeed, the nonnormable spaces of entire functions and the countable product of lines support -linear operators with residual sets of hypercyclic vectors, for .
1. Introduction
The study of linear dynamics has attracted the interest of a number of researchers from different areas over the past decades. Despite the several isolated examples in the literature due to Birkhoff [1], MacLane [2], and Rolewicz [3], it was not until the eighties with the unpublished Ph.D. thesis of Kitai [4] and the papers by Beauzamy [5] and by Gethner and Shapiro [6] when the notion of hypercyclicity started to become popular among mathematicians devoted to operator theory and functional analysis. This interest was fostered by the extension of the definition of chaos in the sense of Devaney to the linear setting by Godefroy and Shapiro [7]. The state of the art on linear dynamics was first described by Grosse-Erdmann in [8] and revisited in [9]; see also [10]. Evidences of the area’s maturity are the recent monographs of Bayart and Matheron [11] and of Grosse-Erdmann and Peris-Manguillot [12].
Throughout this paper, denotes an infinite-dimensional separable Fréchet space over the real or complex scalar field and denorbtes the space of linear and continuous operators on , endowed with the topology of uniform convergence over bounded sets. We recall that a linear operator is said to be hypercyclic if there exists some such that the orbit is dense in . This notion is equivalent to transitivity in the linear setting by the Birkhoff Transitivity Theorem; see for instance [12, Theorem 1.16].
The notion of hypercyclicity is related to the ones of supercyclicity and cyclicity that appeared in connection with the invariant subspace problem; see for instance [13, 14]. We recall that an operator is said to be cyclic if there is some such that its orbit has dense linear span (i.e., ) and it is said to be supercyclic if there exists a vector such that the set of scalar multiples of the orbit is dense (i.e., ).
Apart from these, several notions—either adapted from other areas to the linear setting or conceived within linear dynamics themselves—have appeared in recent years to describe the dynamic behaviour of a linear operator: weak mixing [15, 16], frequent hypercyclicity [17], disjoint hypercyclicity [18, 19], distributional chaos [20], the specification property [21], irregular vectors [22], Li Yorke chaos and distributionally irregular vectors [23], and bihypercyclicity [24], among others.
The notion of bihypercyclicity, due to Grosse-Erdmann and Kim, was introduced to extend the notion of hypercyclicity from operators to bilinear operators. Let us recall their notion of orbit for a given bilinear operator . Given , we define the sequence of sets as follows: So and so on. Figure 1 shows an organized scheme for computing the items in , , and . Then, the orbit of under , in the sense of Grosse-Erdmann and Kim, is defined as The bilinear operator is said to be bihypercyclic if there exists a pair whose orbit is dense in . If so, is called a bihypercyclic vector for .
Under this definition, the authors in [24] proved several interesting results such as the lack of density of the set of bihypercyclic vectors of for a bihypercyclic operator [24, Theorem 1], the bihypercyclicity of nonzero scalar multiples of a bihypercyclic operator [24, Theorem 2], and the existence of bihypercyclic operators on finite-dimensional spaces [24, Theorem 10] and on any separable Banach space [24, Theorem 11]. They also provided a general method for constructing bihypercyclic bilinear operators [24, Proposition 4]; if one can find a vector such that the operator is hypercyclic on , then is bihypercyclic on . This follows by noting that the orbit of a vector under lies inside the orbit of under .
Nevertheless, the notion of iterating an -linear mapping is not evident when and other interpretations may be worth considering. Our attempt is to consider an orbit of a pair under a bilinear operator in such a way that the iterates can be arranged sequentially, and not in a network shape as Figure 1 shows.
In what remains, we consider the following notion of orbit for an -linear operator, inspired in difference equations. Whereas each state of a discrete dynamical system given by an operator is determined by one preceding state (i.e., ), a state in a system given by an -linear operator relies on preceding states (i.e., ).
Definition 1 (orbit under an -linear operator). Let be an -linear operator, where . Each -tuple determines a unique sequence satisfyin
We say that is the -linear orbit for with initial conditions and denote it .
For the case of a bilinear operator, this definition of orbit is simpler than the one used for bihypercyclicity, thanks to the linear order in computing the “iterates” of the initial conditions, as Figure 2 shows.
With this new type of orbit, it is natural to consider the following definition.
Definition 2. One says that an -linear operator is hypercyclic if there exists an -tuple whose orbit (in the sense of Definition 1) is dense in . If is dense, we say that is supercyclic. Such a vector is said to be hypercyclic or supercyclic for , respectively.
Definition 1 provides the following connection with the theory of universal sequences.
Remark 3. Given a continuous -linear map , consider the sequence of continuous maps inductively defined as follows: for , is the projection of the th coordinate of onto . For and , we let Then, the orbit of a vector under is precisely the “orbit” of under the action of . That is, In particular, is hypercyclic for if and only if it is universal for the sequence in and a similar observation holds for the supercyclic case.
Given that the set of universal vectors for a given sequence in is either residual or not dense [8, Proposition 6], we immediately have the following consequence.
Proposition 4. Let be a hypercyclic -linear operator on a Fréchet space , where . Then the set of hypercyclic vectors for is either residual in or not dense in .
On Section 2, we show that every separable infinite-dimensional Fréchet space supports a supercyclic -linear operator, for any (Theorem 5). On Section 3, we show that the space , the countably infinite product of lines, supports hypercyclic -linear operators, for any . We also show that in contrast with the set of bihypercyclic vectors being a nondense set [24, Theorem 1], the set of hypercyclic vectors for an -linear operator can be residual on . On Section 4, we show that the space of entire functions—which unlike supports continuous norms—does support hypercyclic -linear operators, for .
2. Existence of Supercyclic -Linear Operators
Theorem 5. Every separable infinite-dimensional Fréchet space supports, for each , an -linear operator having a residual set of supercyclic vectors.
The proof of Theorem 5 makes use of the following result by Bonet and Peris, which has been a key ingredient to prove the existence of hypercyclic operators on Fréchet spaces different from .
Lemma 6 (see [25, Lemma 2]). Let be a separable infinite dimensional Fréchet space . There are sequences and such that(1) converges to in and is dense in ;(2) is -equicontinuous in ;(3) if and for all .
With this notation, Bonet and Peris proved that the operator resulted in being hypercyclic on . Lemma 6 can be compared with the well-known result by Ovsepian and Pełczyńsky on the existence of a fundamental total and bounded pair of biorthogonal sequences on separable Banach spaces [26]. This last result was used by Herzog to show that every infinite-dimensional separable Banach space supports a supercyclic operator [27], which was of the form where and were a pair of biorthogonal sequences given by the Ovsepian and Pełczyńsky result. This operator is in fact a generalized backward shift operator [7]. In [28], Salas extended to these types of operators previous results, due to Hilden and Wallen, on the supercyclicity of unilateral backward weighted shifts on spaces [14]. The supercyclicity of generalized backward shift operators can be characterized in terms of having dense range or verifying the supercyclicity criterion [29].
Using the operator , but in this case using the sequences given by Bonet and Peris lemma instead of the ones by Ovsepian and Pełczyńsky, and taking again the tensor product approach, we can prove that every separable infinite dimensional Fréchet space supports an -supercyclic operator. We point out that the case will be established once we show Example 9.
We also use the following lemma, due to Grosse-Erdmann [30].
Lemma 7. Let be a sequence of continuous mappings , , where and are topological vector spaces. If is a Baire space and is metrizable, then the set of universal vectors for is residual in if and only if the set is dense in .
Proof of Theorem 5. Consider the -linear operator , where is fixed. That is, is given by
Notice that, for any given vectors in , the orbit
satisfies
with the scalar depending only on the scalars in the set
where if and only if . Notice also that for any of the form with and we have
where and for each and for each . Now, let , and be vectors in the linear span of and let be given. By Remark 3 and Lemma 7, it suffices to show there exist vectors in , a scalar , and a positive integer so that
where and are the metrics on and , respectively, and is the sequence in associated with by relation (6).
Without loss of generality, we may assume that and that with for each and for some . Consider the vectors for and
where is chosen small enough so that . Then
and by (14) the scalar is nonzero, as
So (16) is satisfied taking and .
3. Hypercyclic -Linear Operators on
Let us consider the space endowed with the product topology. This can be given either by the metric or by the family of continuous seminorms defined as
Let denote a countable dense subset of satisfying if and only if . Also, denotes the unweighted backward shift operator defined over a sequence of numbers as
The hypercyclicity phenomenon on has been already considered as a particular case of Fréchet spaces, since it is the furthest Fréchet space from having a continuous norm; see, for instance, [25, 31–33]. The main result of this section is the following.
Theorem 8. For each integer , there exists an -linear operator on that supports a dense -linear orbit.
We show Theorem 8 by providing two examples of such operators. The first one is constructed as a tensor product like [24, Example 1].
Example 9. Let be fixed and consider the -linear operator on , where is the first coordinate functional on . That is, is given by
Then, is hypercyclic. To see this, notice that for any vector in the orbit
is the following: , , , and for each we have
where
and where is the Fibonacci sequence of order and seed
That is, is recursively defined by
We now construct so that is dense in , as follows: let be a sequence of integers so that and for all and let be a dense sequence in so that each satisfies if and only if .
We define the first coordinates of by
and for we let be defined as in (26).
Next, we define
and for we again let be defined by (26). Inductively, having defined and for , we define and for as
and is defined as in (26). So, we have defined in of the form
Finally, to prove the denseness of we take an arbitrary in and . Let be large enough so that and . Then,
since by (25) the first coordinates of are all zero.
Remark 10. We note that the set of hypercyclic vectors for the operator of the previous example is residual in .
Proof of Remark 10. By Proposition 4, it suffices to verify that is dense in . The orbit of a given in is given by , , and for
where
and where each is an -Fibonacci sequence as follows.
For ,
For ,
Let in be given and . We want to find some in with
Let be large enough so that . Perturbing each if necessary, we may assume without loss of generality that for and that with if and only if . Let denote a countable dense subset of satisfying if and only if and let be a sequence of positive integers satisfying . Consider the vectors for and
where the coordinates of are to be determined. Notice that by our selection of regardless of how the ’s are chosen. Now each scalar in (34) depends only on
and every scalar will be nonzero as long as each of the ’s are nonzero. In particular, by (34) we can define such so that
So is in and the conclusion follows.
We next provide another example of a hypercyclic -linear operator on , avoiding the tensor product technique. In Example 11, each coordinate of an element in is used to conform the iteration under the multilinear operator . This allows us to provide a much simpler expression of some initial conditions that yield a dense orbit under . We point out that since roots of different orders are taken, we only consider this example on .
Example 11. Let be fixed and consider the -linear operator , where , given by Let be a sequence of integers so that and for all and let denote the Fibonacci sequence of integers recursively defined by Taking again the set , we consider the vector in given by That is, is of the form where in each case denotes any of the roots of . By (42), the -linear orbit of the initial conditions under is given by which is dense in . This last part can be proved in a similar way as we did with Example 9.
4. Hypercyclic -Linear Operators on
One may wonder whether lacking a continuous norm, such as does, is a requirement for a space to support hypercyclic -linear operators. We answer this in the negative, with the following.
Theorem 12. The space supports a hypercyclic bilinear operator.
The proof of Theorem 12 relies on Lemmas 13 and 14. First, let us introduce some notation. Let us define the antiderivative operator on as for every entire function . It is clear that, on the monomials , , this operator returns and that in . Thus, the equicontinuity of any finite collection of iterates of the derivative operator on immediately gives the following.
Lemma 13. Let be a complex polynomial. Let and let ; there is some such that if and , then
If is a Fibonacci sequence of order (i.e., for where are given), then where is the real solution of that is closest to 2. Hence, by Lemma 13 we have the following.
Lemma 14. Let be a complex polynomial and let be the Fibonacci sequence defined in (28). Let be a finite sequence of complex numbers satisfying where is the real solution of that is closest to . Let also , , and . Then, there exists some such that for all for all .
Proof of Theorem 12. Consider the bilinear operator ,
where is the evaluation at and is the operator of complex differentiation on . We seek a hypercyclic vector for of the form , with . Notice that for any the orbit is given by and
where
and where is the Fibonacci sequence in (28) given by and for . The construction of the function is inspired by the construction of a hypercyclic function for the derivative operator in [34]. We first consider a dense sequence of polynomials
satisfying, for each that(a), and , and(b), where for each the scalar associated to is defined as
(It is simple to see that such exists: first get a dense sequence and perturb if necessary the coefficients of each by at most so that . The resulting sequence, which we call , is also dense. For each polynomial in , the polynomial satisfies . Then results after reordering .) We also consider another dense family of functions defined from the polynomials in as
Several Fibonacci sequences will be defined at each step of the inductive construction of , and so that denotes the power of the coefficient of that appears in the factor in the -th iterate of the orbit of given by (53). All these Fibonacci sequences are of order , and thus we only define in the proof the first two non-zero terms of each such sequences. We also introduce the following notation:
Clearly, the following relation holds
Now, we are ready to construct .
Step 1. Let , and define .
Step 2. We first let and denote the Fibonacci sequences whose first non-zero terms are for and . Next we define , where is chosen so that (2.1) , (2.2) for all , (2.3) for all , and (2.4) for all .
Conditions (2.2) and (2.3) are obtained thanks to (48) and the fact that , and Condition (2.4) is obtained from Lemma 14. We finish this step defining . Notice that for any the orbit
satisfies that , where for we let ().
Step 3. We let and denote the Fibonacci sequences whose first non-zero terms are given by
and define , where satisfies (3.1) , (3.2) for all . (3.3) for all . (3.4) , for all .
Condition (3.2) holds again thanks to (48). To see why Condition (3.3) can be obtained, recall that thanks to the order in which the polynomials and appear in the dense sequence given in (54). Notice also that the quotients are eventually constant and coincide with , so that coincides with and thus eventually by (2.2). On the other hand for all by Condition (3.2). All this together with (58) allow us to obtain (3.3), since by construction. Condition (3.4) can now be obtained as a combination of Lemma 13 and (3.3). Finally, we let , and as in Step 2 for any the orbit
satisfies that .
Step 4. We define the Fibonacci sequences needed in this step, with . The first two non-zero terms of each sequence are given by
We set , where is chosen to satisfy the following: (4.1) , (4.2) for all , (4.3) for all , and (4.4) , for all .
Condition (4.2) is given by (48). Let us proceed to check condition (4.3). The idea is very similar to the one used for condition (3.3). By the definition of the Fibonacci sequences in this step, the quotients are constant for every and coincide with . Therefore for every . Analogously, is constant for every and coincides with . This gives . Again, since by construction, then for all . In addition, by condition (4.2) since is greater than by definition. Combining these estimations we get (4.3). Lastly, condition (4.4) can be obtained from Lemma 13 using condition (4.3).
We also take so that, as in Steps 2 and 3, for any the orbit
satisfies that .
Step n. Let us assume that we have done the previous steps. Let us start with the definition of the Fibonacci sequences, . The first non-zero terms of each sequence are the following ones:
We define , where satisfies: (n.1) , (n.2.1) if is odd, then for all , (n.2.2) if is even, then for all , (n.3) for all , and (n.4) , for all .
Both conditions (n.2.1) and (n.2.2) can be deduced using the formula (48). For proving condition , we first prove that for every odd number and . This is due to the general fact that the quotients are constant for all and for all , since each one coincides with , respectively. So that coincides with . By the previous steps if is odd and if is even. Taking this into account and since , then for all large enough.
Now, if is even, then condition (n.3) is obtained applying the previous argument with all the odd numbers . If is odd, we apply the previous argument to all odd numbers and we have still to show that for all large enough, but this holds by condition (n.2.2) and (58) since by definition. Combining these estimations we get (n.3). As before, condition (n.4) can be obtained from Lemma 13 using condition (n.3).
We finish this step defining . Now, for any the orbit
satisfies that .
To sum up, our function will be defined as . Clearly, it is well defined for every and converges uniformly on bounded sets of , because of statements and the fact that the sum is bounded in modulus by .
Now, take an arbitrary function and , . We choose such that and . Then we take such that(i),(ii) and(iii).
We will see that . This holds using the aforementioned estimations and the statements .
Remark 15. The map , , is continuous, injective, of dense range, and satisfies that , where is the backward shift on and the derivative operator on . So for any the image of under is the orbit . In particular, is hypercyclic for whenever is hypercyclic for , what together with the example constructed to show Theorem 12 gives another proof of the hypercyclicity of the -linear operator in Example 9 for the case .
5. Final Comments
We have not come up with examples of hypercyclic -linear operators on Banach spaces. Hence, we pose the following.
Problem 16. Does any Banach space support a hypercyclic -linear operator, for some ?
We note that if is an -linear operator with and being a Banach space, the set of hypercyclic vectors for must be nondense in . Indeed, the continuity of together with the fact that ensures that any in with satisfies that the orbit converges to zero, where We note that, by the above argument, a Banach space even lacks hypercyclic subspaces of dimension one for any -linear operator with .
In view of Proposition 4 and Remark 10, it is also natural to ask.
Problem 17. Must the set of hypercyclic vectors of an -linear operator be dense in ?
Of course, an affirmative answer to Problem 16 gives a negative answer to Problem 17 and an affirmative answer to Problem 17 gives a negative answer to Problem 16.
To conclude, we note that it is natural to seek extensions of results and notions in linear dynamics to -linear dynamics. For instance, we may propose the following notion of chaos for -linear operators, motivated by the notion of Devaney-chaos in linear dynamics [35].
Definition 18. A vector is said to have a periodic orbit under an -linear operator whenever there exist some and some such that for all . Such a vector is said to be -periodic for . We say that is -linear Devaney chaotic if it is a hypercyclic -linear operator and it has a dense set of periodic orbits.
We note next that the -linear operators of Examples 9 and 11 are also Devaney chaotic. Concerning Example 9, consider a dense sequence of eventually null elements of , namely, with if and only if . Then, for every element we define the corresponding vector where is defined in (26). Let us also define the elements It follows that the initial conditions force to be an -periodic point for the operator defined on (23). Finally, the vectors are dense because of the denseness of .
Let us proceed with Example 11. Consider a dense sequence of eventually null elements of , namely, with if and only if . Then, for every element , we define the corresponding vector where denotes one of the roots of . It follows that the initial conditions force to be an -periodic point for the operator defined on (42). To sum up, the set of periodic points for is dense in .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are supported in part by MEC and FEDER, Project MTM2010-14909. The second author also wishes to thank Programa de Investigación y Desarrollo de la UPV, Referencia SP20120700. They also thank the referees for their very helpful comments.