On a Chaotic Weighted Shift of Order in Bargmann Space
This paper is devoted to the study of the chaotic properties of some specific backward shift unbounded operators ; realized as differential operators in Bargmann space, where and are the standard Bose annihilation and creation operators such that .
It is well known that linear operators in finite-dimensional linear spaces cannot be chaotic but the nonlinear operator may be. Only in infinite-dimensional linear spaces can linear operators have chaotic properties. These last properties are based on the phenomenon of hypercyclicity or the phenomen of nonwandercity.
The study of the phenomenon of hypercyclicity originates in the papers by Birkoff  and Maclane  who show, respectively, that the operators of translation and differentiation, acting on the space of entire functions, are hypercyclic.
The theories of hypercyclic operators and chaotic operators have been intensively developed for bounded linear operators; we refer to [1, 3–5] and references therein. For a bounded operator, Ansari asserts in  that powers of a hypercyclic bounded operator are also hypercyclic.
For an unbounded operator, Salas exhibits in  an unbounded hypercyclic operator whose square is not hypercyclic. The result of Salas shows that one must be careful in the formal manipulation of operators with restricted domains. For such operators, it is often more convenient to work with vectors rather than with operators themselves.
Now, let be an unbounded operator on a separable infinite dimensional Banach space . A point is called wandering if there exists an open set containing such that for some and for all one has . (In other words, the neighbourhood eventually never returns). A point is called nonwandering if it is not wandering.
A closed subspace has hyperbolic structure if: , , and , where (the unstable subspace) and (the stable subspace) are closed. In addition, there exist constants and , such that:
(i) for any ,
(ii) for any .
is said to be a nonwandering operator relative to which has hyperbolic structure if the set of periodic points of is dense in .
For the nonwandering operators, they are new linear chaotic operators. They are relative to hypercyclic operators, but different from them in the sense that some hypercyclic operators are not non-wandering operators and there also exists a non-wandering operator, which does not belong to hypercyclic operators (see , Remark 3.5). In fact, suppose is a bounded linear operator and is invertible; if is a hypercyclic operator, then (see , Remark 4.3) but if is a non-wandering operator, then where is the unit circle.
Now, when a linear operator is not invertible, there exist operators which are not only non-wandering but also hypercyclic. Recently, these theories began to be developed on some concrete examples of unbounded linear operators; see [1, 10–12]. On the basis of the work in , we study the phenomenons of chaoticity of some specific backward shift unbounded operators realized as differential operators in Bargmann space  (the space of entire functions with Gaussian measure), where and are the standard Boson annihilation and creation operators satisfying the commutation relation
Of special interest is a representation of these operators and as linear operators in a separable Hilbert spanned by eigenvectors of the positive semidefinite number operator .
One has the well-known relations We denote the Bargmann space  by
The scalar product on is defined by and the associated norm is denoted by .
An orthonormal basis of is given by is closed in , where the measure and is closed related to by an unitary transform of onto given in  by the following integral transform:
If the integral converges absolutely.
In this Bargmann representation, the annihilator and creator operators are defined by
Now, we define
An orthonormal basis of is given by
Hence, a family of weighted shifts is defined as follows:
Remark 1.1. (i) For , the operator is the derivation in Bargmann space, and it is the celebrated quantum annihilation operator.
(ii) is a weighted shift with weight for .
(iii) It is known that with its domain is a chaotic operator in Bargmann space.
(iv) for all where and .
(v) The function is called a coherent normalized quantum optics (see [15, 16]).
Remark 1.2. (i) For , the operator has as adjoint the operator .
(ii) is a weighted shift with weight for and it is known that is a not self-adjoint operator and is chaotic in Bargmann space . This operator plays an essential role in Reggeon field theory (see [17, 18]).
(iii) The operators arising also in the Jaynes-Cummings interaction models, see for example a model introduced by Obada and Abd Al-Kader in , the interaction Hamiltonian for the model is where are the Rabi frequencies and and are the Lamb-Dicke; = 1,2. The operators and act on the ground state and excited state as follows: and .
(iv) On , which is the orthogonal of span in Bargmann space, the adjoint of is such that
This paper is organized as follows: in Section 2, we recall the definition of the chaoticity for an unbounded operator following Devaney and sufficient conditions on hypercyclicity of unbounded operators given by Bés-Chan-Seubert theorem . As our operator is an unilateral weighted backward shift with an explicit weight, we use the results of Bés et al. to proof the chaoticity of in Bargmann space (we can also use the results of Bermúdez et al.  to proof the chaoticity of our operator ). Then, we construct the hyperbolic structure associated to . In the appendix, we present a direct proof of the chaoticity of based on the Baire Category theorem. The last theorem is essential to proof that the operator is topologically transitive and can be used for interested reader.
2. Chaoticity of the Operator on
Definition 2.1. Let be an unbounded linear operator on a separable infinite dimensional Banach with domain dense in and such that is closed for all positive integers .(a)The operator is hypercyclic if there exists a vector such that and if the orbit is dense in . The vector is called a hypercyclic vector of .(b)A vector is called a periodic point of if there exists such that . The operators having both dense sets of periodic points and hypercyclic vectors are said to be chaotic following the definition of Devaney [20, 21].
Sufficient conditions for the hypercyclicity of an unbounded operator are given in the following Bés-Chan-Seubert theorem:
Theorem 2.2 (Bés-Chan-Seubert ). Let be a separable infinite dimensional Banach, and let be a densely defined linear operator on . Then, is hypercyclic if(i) is a closed operator for all positive integers m,(ii)there exists a dense subset of the domain of and a (possibly nonlinear and discontinuous) mapping so that is the identity on and pointwise on F as
Theorem 2.3. Let be the Bargmann space with orthonormal basis . Let with domain , where . Then, is a chaotic operator.
Remark 2.4. (i) Following the ideas of Gross-Erdmann in [4, 5] or the Theorem 2.4 of Bermúdez et al. , we can use a test on the weight of to give a proof of the chaoticity of .
We choose to give a proof under lemma form based on the theorem of Bès et al. recalled above, we also indicate in the appendix the utilization of the Baire category theorem in the hepercyclicity theory and we prove that possesses a certain "sensitivity to initial conditions" though this property is redundant in Devaney's definition (see Banks et al. in ).
(ii) Let be an unbounded operator on separable infinite dimensional Banach . It may happen that vector f , but Tf fails to be in the domain of . We can exhibit a closed operator whose square is not. For example, the operator acting on defined by with domain , where is the derivative of and is a function in with , where is the classical Sobolev space. Then , is a closed operator and , where is the domain of but the operator is not closed and has not closed extension.This operator can, for example, justify the asumption (a) of the Definition 2.1 for the unbounded linear operators.
Lemma 2.5. For each positive integer , the operator , with domain , is a closed operator.
Proof. As is closed if and only if the graph is a closed linear manifold of , then let be a sequence in which converges to in . As converges to in , then converges to pointwise on and converges to pointwise on . As converges to , we deduce that and , hence is closed.
Lemma 2.6. Let with domain , where , , and for . Then, is hypercyclic.
Proof. Let . This space is dense in .
Let : and .
Then, , that is, .
Now, as for all we deduce that any element of can be annihilated by a finite power of since as when , we have Now, the hypercyclicity of follows from the theorem of Bés et al. recalled above.
Lemma 2.7. Let with domain , where , , and for . Then, there exist and such that .
Proof. Let and
then is in the domain of and it is an eigenvector for corresponding to eigenvalue , therefore it is a periodic point of . is a root of unity.
In fact, let and , then as there exist and such that since for , we have and is in Bargmann space. Now as, we get We deduce that
that is, .
Thus, we get
Therefore, is the eigenvector corresponding to the eigenvalue and is a periodic point of , where is a root of unity.
Lemma 2.8. The set of periodic points of is dense in .
is dense in , otherwise there exists nonzero vector which is orthogonal to .
is a continuous function on the closed unit disc which is holomorphic on the interior and vanishes at each root of unity, hence on the entire unit circle, hence vanishes for all . We deduce that for , then is dense in .
Definition 2.10. Let be an unbounded linear operator on a separable infinite dimensional Banach whose domain is dense in , and let be closed for all positive integers .(a)A closed subspace has hyperbolic structure if , , and , where (the unstable subspace) and (the stable subspace) are closed. In addition, there exist constants and , such that:(i)For any (the vectors of are exponentially expanded, we say they belong to the unstable subspace ).(ii)For any (some vectors are contracted exponentially fast by the iterates of the operator , we say they belong to the stable subspace ).(b)If there exists a closed subspace which has hyperbolic structure relative to and the set of periodic points of is dense in , then is said to be a nonwandering operator relative to following the definition of Tian et al. .
Since is chaotic operator on , so has dense set of periodic points on , we only need to construct an hyperbolic structure associated to it in to obtain:
Theorem 2.11. Let be the Bargmann space with orthonormal basis .
Let with domain , where .
Then, is a nonwandering operator.
Proof. We construct a closed invariant subspace such that has hyperbolic structure.
For , the function defined by (2.2).
is in the domain of and is an eigenvector for corresponding to the eigenvalue .
Let , , and , where represents direct sum.
We will verify that has an hyperbolic structure.
For , there exists a sequence such that And for each positive integer , we have
Next, we will prove is the invariant subspace of .
Let , then where.
Now for , then there exists , such that , where . Therefore, .
Similarly, let , we deduce that and if we chose , then we have Then, has hyperbolic structure and is nonwandering operator relative to .
Here, the linear space is formed by the (spectral) subspace corresponding to the eigenvalues of of modulus less than 1, while the unstable subspace corresponds to those of modulus greater than 1.
We conclude that main results of this work can be considered in  as an introduction to study of the operators with and particularly, to study the chaoticity of .
AppendixLet us recall below the essential spaces and operators used in above sections(i),(ii),(iii) with and for , (iv),(v).
Then, we have the following.
Lemma A.1. For arbitrary , there exists such that and .
Proof. As is dense in , then for arbitrary , there exists such that .
Let m a natural number, as for , then we get Now, for arbitrary and , , we have
As for , we have , then hence tends pointwise to zero on .
By choosing diagonal element of , we get As on , then we can write , that is,
Now as , where , is the kernel space of then , is dense in and for arbitrary , there exists such that . Therefore, tends pointwise to zero on a dense subset of .
For arbitrary , there exists such that , therefore Particularly, Let Then, .
Lemma A.2. Let , where , is an enumeration of open ball in with centers in a countable dense subset of , then is dense in .
Proof. The above lemma imply for arbitrary and that there exists such that , and , hence is dense in and Baire category theorem implies is dense in . Hence, is topologically transitive.
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