On a Chaotic Weighted Shift 𝑧^𝑝𝑑^𝑝+𝟏/𝑑𝑧^𝑝+𝟏 of Order 𝑝 in Bargmann Space

Abstract

This paper is devoted to the study of the chaotic properties of some specific backward shift unbounded operators 𝐻_𝑝=𝐴^∗𝑝𝐴^𝑃+1; 𝑝=0,1,… realized as differential operators in Bargmann space, where 𝐴 and 𝐴^∗ are the standard Bose annihilation and creation operators such that [𝐴,𝐴^∗]=𝐼.

1. Introduction

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 [1] and Maclane [2] 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 [6] that powers of a hypercyclic bounded operator are also hypercyclic.

For an unbounded operator, Salas exhibits in [7] 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 𝑇 be an unbounded operator on a separable infinite dimensional Banach space 𝑋. A point 𝜙 is called wandering if there exists an open set 𝑈 containing 𝜙 such that for some 𝑛₀<∞ and for all 𝑛>𝑛₀ one has 𝑇^𝑛(𝑈)⋂𝑈=∅. (In other words, the neighbourhood eventually never returns). A point 𝜙 is called nonwandering if it is not wandering.

A closed subspace 𝐸⊂𝑋 has hyperbolic structure if: 𝐸=𝐸^𝑢⊕𝐸^𝑠, 𝑇𝐸^𝑢=𝐸^𝑢, and 𝑇𝐸^𝑠=𝐸^𝑠, where 𝐸^𝑢 (the unstable subspace) and 𝐸^𝑠 (the stable subspace) are closed. In addition, there exist constants 𝜏(0<𝜏<1) and 𝐶>0, such that:

(i) for any Φ∈𝐸^𝑢,𝑘∈ℕ,𝐶𝜏^−𝑘||Φ||≤||𝑇^𝑘Φ||,

(ii) for any Ψ∈𝐸^𝑠,𝑘∈ℕ,||𝑇^𝑘Ψ||≤𝐶𝜏^𝑘||Ψ||.

𝑇 is said to be a nonwandering operator relative to 𝐸 which has hyperbolic structure if the set of periodic points of 𝑇 is dense in 𝐸.

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 [8], Remark   3.5). In fact, suppose 𝑇 is a bounded linear operator and 𝑇 is invertible; if 𝑇 is a hypercyclic operator, then 𝜎(𝑇)⋂𝜕𝐷≠Φ (see [9], Remark   4.3) but if 𝑇 is a non-wandering operator, then 𝜎(𝑇)⋂𝜕𝐷=Φ where 𝜕𝐷 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 [13], we study the phenomenons of chaoticity of some specific backward shift unbounded operators 𝐻_𝑝=𝐴^∗𝑝𝐴^𝑝+1=𝑧^𝑝(𝑑^𝑝+1/𝑑𝑧^𝑝+1);𝑝=0,1,… realized as differential operators in Bargmann space [14] (the space of entire functions with Gaussian measure), where 𝐴 and 𝐴^∗ are the standard Boson annihilation and creation operators satisfying the commutation relation 𝐴,𝐴^∗=𝐼. (1.1)

Of special interest is a representation of these operators 𝐴 and 𝐴^∗ as linear operators in a separable Hilbert spanned by eigenvectors |𝑛⟩;𝑛=0,1,… of the positive semidefinite number operator 𝑁=𝐴^∗𝐴.

One has the well-known relations𝐴|𝑛⟩=√𝑛|𝑛−1⟩,𝐴^∗|𝑛⟩=√𝑛+1|𝑛+1⟩. (1.2) We denote the Bargmann space [14] by 𝐵=𝜙∶ℂ⟶ℂentire,_ℂ𝜙(𝑧)²𝑒^−|𝑧|2𝑑𝑥𝑑𝑦<∞. (1.3)

The scalar product on 𝐵 is defined by𝜙,𝜓=_ℂ𝜙(𝑧)𝜓(𝑧)𝑒^−|𝑧|2𝑑𝑥𝑑𝑦, (1.4) and the associated norm is denoted by ‖⋅‖.

An orthonormal basis of 𝐵 is given by𝑒_𝑛(𝑧)=𝑧^𝑛√𝑛!;𝑛=0,1,…. (1.5)𝐵 is closed in 𝐿₂(ℂ,𝑑𝜇(𝑧)), where the measure 𝑑𝜇(𝑧)=𝑒^−|𝑧|2𝑑𝑥𝑑𝑦 and is closed related to 𝐿₂(ℝ) by an unitary transform of 𝐿₂(ℝ) onto 𝐵 given in [14] by the following integral transform:𝜙(𝑧)=_ℝ𝑒^{−(1/2)(𝑧2+𝑢2√)+2𝑢𝑧}𝑓(𝑢)𝑑𝑢. (1.6)

If 𝑓∈𝐿₂(ℝ) the integral converges absolutely.

In this Bargmann representation, the annihilator and creator operators are defined by(𝑧)=𝑑𝐴𝜙𝜙(𝑧)𝑑𝑧withdomain𝐷(𝐴)=𝑑𝜙∈𝐵;,𝐴𝑑𝑧𝜙∈𝐵_∗𝜙(𝑧)=𝑧𝜙(𝑧)withdomain𝐷𝐴^∗=𝜙∈𝐵;𝑧𝜙∈𝐵. (1.7)

Now, we define 𝐵_𝑝=𝑑𝜙∈𝐵;^𝑗𝑑𝑧^𝑗𝜙(0)=0,0≤𝑗≤𝑝. (1.8)

An orthonormal basis of 𝐵_𝑝 is given by𝑒_𝑛(𝑧)=𝑧^𝑛√𝑛!;𝑛=𝑝+1,𝑝+2,…. (1.9)

Hence, a family of weighted shifts 𝐻_𝑝 is defined as follows:𝐻_𝑝=𝐴^∗𝑝𝐴^𝑝+1withdomain𝐷𝐻_𝑝=𝜙∈𝐵;𝐻_𝑝𝜙∈𝐵𝐵_𝑝. (1.10)

Remark 1.1. (i) For 𝑝=0, the operator 𝐻₀=𝐴 is the derivation in Bargmann space, and it is the celebrated quantum annihilation operator.
(ii) 𝐻^∗₀𝑒_𝑛=√𝑛+1𝑒_𝑛+1 is a weighted shift with weight 𝜔_𝑛=√𝑛+1 for 𝑛=0,1,….
(iii) It is known that 𝐻₀ with its domain 𝐷(𝐻₀) is a chaotic operator in Bargmann space.
(iv) 𝐻₀𝜙_𝜆(𝑧)=𝜆𝜙_𝜆(𝑧) for all 𝜆∈ℂ where 𝜙_𝜆(𝑧)=∑^∞_𝑛=0(𝜆^𝑛/√𝑛!)𝑒_𝑛(𝑧) and ‖𝜙_𝜆‖²=𝑒^|𝜆|2.
(v) The function 𝑒^−|𝜆|2𝜙_𝜆(𝑧) is called a coherent normalized quantum optics (see [15, 16]).

Remark 1.2. (i) For 𝑝=1, the operator 𝐻₁=𝐴^∗𝐴²=𝑧(𝑑²/𝑑𝑧²) has as adjoint the operator 𝐻^∗₁=𝑧²(𝑑/𝑑𝑧).
(ii) 𝐻^∗₁𝑒_𝑛=𝑛√𝑛+1𝑒_𝑛+1 is a weighted shift with weight 𝜔_𝑛=𝑛√𝑛+1 for 𝑛=1,… and it is known that 𝐻₁+𝐻^∗₁ is a not self-adjoint operator and is chaotic in Bargmann space [13]. This operator plays an essential role in Reggeon field theory (see [17, 18]).
(iii) The operators 𝐻_𝑝 arising also in the Jaynes-Cummings interaction models, see for example a model introduced by Obada and Abd Al-Kader in [19], the interaction Hamiltonian for the model is 𝐻_𝐼=−^∞_𝑝=0Ω₁𝑒^𝑖𝜙1𝑒^{−(𝜂21/2)}𝑖𝜂₁^2𝑝+1!𝐻𝑝!𝑝+1^∗𝑝+Ω₂𝑒^𝑖𝜙2𝑒^{−(𝜂22/2)}𝑖𝜂₂^2𝑝+1!𝐻𝑝!𝑝+1^𝑝𝜎₋+ℎ⋅𝑐, (1.11) where Ω_𝑗 are the Rabi frequencies and and 𝜂²_𝑗 are the Lamb-Dicke; 𝑗 = 1,2. The operators 𝜎₋ and 𝜎₊ act on the ground state |𝑔⟩ and excited state |𝑒⟩ as follows: 𝜎_±|𝑔⟩=(1±1/2)|𝑒⟩ and 𝜎_±|𝑒⟩=(1∓1/2)|𝑔⟩.
(iv) On 𝐵_𝑝,𝑝=0,1,…, which is the orthogonal of span {𝑒_𝑛;𝑛≤𝑝} in Bargmann space, the adjoint of 𝐻_𝑝 is 𝐻^∗_𝑝=𝑧^𝑝+1(𝑑^𝑝/𝑑𝑧^𝑝) such that𝐻^∗_𝑝𝑒_𝑛=𝜔_𝑛𝑒_𝑛+1withweight𝜔_𝑛=√𝑛+1𝑛!𝑛−𝑝!for𝑛≥𝑝≥0. (1.12)
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 [10]. As our operator 𝐻_𝑝 is an unilateral weighted backward shift with an explicit weight, we use the results of Bés et al. to proof the chaoticity of 𝐻_𝑝 in Bargmann space (we can also use the results of Bermúdez et al. [11] to proof the chaoticity of our operator 𝐻_𝑝). Then, we construct the hyperbolic structure associated to 𝐻_𝑝. In the appendix, we present a direct proof of the chaoticity of 𝐻_𝑝 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 𝐻_𝑝=𝑧^𝑝(𝑑^𝑝+1/𝑑𝑧^𝑝+1) on 𝐵_𝑝

Definition 2.1. Let 𝑇 be an unbounded linear operator on a separable infinite dimensional Banach 𝑋 with domain 𝐷(𝑇) dense in 𝑋 and such that 𝑇^𝑛 is closed for all positive integers 𝑛.(a)The operator 𝑇 is hypercyclic if there exists a vector 𝑓∈𝐷(𝑇) such that 𝑇^𝑛𝑓∈𝐷(𝑇) and if the orbit {𝑓,𝑇𝑓,𝑇²𝑓,…} is dense in 𝑋. The vector 𝑓 is called a hypercyclic vector of 𝑇.(b)A vector 𝑔∈𝐷(𝑇) is called a periodic point of 𝑇 if there exists 𝑚 such that 𝑇^𝑚𝑔=𝑔. 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 [10]). Let 𝑋 be a separable infinite dimensional Banach, and let 𝑇 be a densely defined linear operator on 𝑋. Then, 𝑇 is hypercyclic if(i)𝑇^𝑚 is a closed operator for all positive integers m,(ii)there exists a dense subset 𝐹 of the domain 𝐷(𝑇) of 𝑇 and a (possibly nonlinear and discontinuous) mapping 𝑆∶𝐹→𝐹 so that 𝑇𝑆 is the identity on 𝐹 and 𝑇^𝑛,𝑆^𝑛→0 pointwise on F as 𝑛→∞.

Theorem 2.3. Let 𝐵 be the Bargmann space with orthonormal basis 𝑒_𝑛(𝑧)=𝑧^𝑛/√𝑛!. Let 𝐻_𝑝=𝑧^𝑝(𝑑^𝑝+1/𝑑𝑧^𝑝+1)=𝐴^∗𝑝𝐴^𝑝+1 with domain 𝐷(𝐻_𝑝)={𝜙∈𝐵;𝐻_𝑝𝜙∈𝐵}⋂𝐵_𝑝, where 𝐵_𝑝={𝜙∈𝐵;(𝑑^𝑗/𝑑𝑧^𝑗)𝜙(0)=0,0≤𝑗≤𝑝}. Then, 𝐻_𝑝 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. [11], we can use a test on the weight of 𝐻_𝑝 to give a proof of the chaoticity of 𝐻_𝑝.
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 𝐻_𝑝 possesses a certain "sensitivity to initial conditions" though this property is redundant in Devaney's definition (see Banks et al. in [20]).
(ii) Let 𝑇 be an unbounded operator on separable infinite dimensional Banach 𝑋. It may happen that vector f ∈𝐷(𝑇), but Tf fails to be in the domain of 𝑇. We can exhibit a closed operator whose square is not. For example, the operator acting on 𝐿₂(0,1)×𝐿₂(0,1) defined by 𝑇(𝑢,𝑣)(𝑥)=(𝑣^′(𝑥),𝑓(𝑥)𝑣(0)) with domain 𝐷(𝑇)=𝐿₂(0,1)×𝐻₁(0,1), where 𝑣′(𝑥) is the derivative of 𝑣(𝑥) and 𝑓 is a function in 𝐻₁(0,1) with 𝑓(0)=1, where 𝐻₁(0,1) is the classical Sobolev space. Then 𝑇, is a closed operator and 𝐷(𝑇²)=𝐷(𝑇), where 𝐷(𝑇²) is the domain of 𝑇² but the operator 𝑇² 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 𝑚, the operator (𝐻_𝑝)^𝑚, with domain 𝐷((𝐻_𝑝)^𝑚)={𝜙∈𝐵;(𝐻_𝑝)^𝑚𝜙∈𝐵}⋂𝐵_𝑝, is a closed operator.

Proof. As (𝐻_𝑝)^𝑚 is closed if and only if the graph 𝐺((𝐻_𝑝)^𝑚) is a closed linear manifold of 𝐵_𝑝×𝐵_𝑝, then let (𝜙_𝑛,(𝐻_𝑝)^𝑚𝜙_𝑛) be a sequence in 𝐺((𝐻_𝑝)^𝑚) which converges to (𝜙,𝜓) in 𝐵_𝑝×𝐵_𝑝. As 𝜙_𝑛 converges to 𝜙 in 𝐵_𝑝, then 𝑧^𝑝(𝑑^𝑝+1/𝑑𝑧^𝑝+1)𝜙_𝑛 converges to 𝑧^𝑝(𝑑^𝑝+1/𝑑𝑧^𝑝+1)𝜙 pointwise on ℂand (𝐻_𝑝)^𝑚𝜙_𝑛 converges to (𝐻_𝑝)^𝑚𝜙 pointwise on ℂ. As (𝐻_𝑝)^𝑚𝜙_𝑛 converges to 𝜓, we deduce that (𝐻_𝑝)^𝑚)𝜙=𝜓 and 𝜙∈𝐷((𝐻_𝑝)^𝑚), hence 𝐺((𝐻_𝑝)^𝑚) is closed.

Lemma 2.6. Let 𝐻_𝑝=𝑧^𝑝(𝑑^𝑝+1/𝑑𝑧^𝑝+1) with domain 𝐷(𝐻_𝑝)={𝜙∈𝐵;𝐻_𝑝𝜙∈𝐵}⋂𝐵_𝑝, where 𝐻_𝑝𝑒_𝑛=𝜔_𝑛−1𝑒_𝑛−1, 𝑒_𝑛(𝑧)=𝑧^𝑛/√𝑛!, and 𝜔_𝑛=√𝑛+1(𝑛!/(𝑛−𝑝)!) for 𝑛≥𝑝≥0. Then, 𝐻_𝑝 is hypercyclic.

Proof. Let 𝐹={𝜙_𝑘(𝑧)=∑^𝑘_𝑛=𝑝𝑎_𝑛𝑒_𝑛(𝑧)}. This space is dense in 𝐵_𝑝.
Let 𝑆_𝑝: 𝐹→𝐹 and 𝑆_𝑝𝑒_𝑛=(1/𝜔_𝑛)𝑒_𝑛+1;𝑛≥𝑝≥0.
Then, 𝐻_𝑝𝑆_𝑝𝜙_𝑘(𝑧)=𝜙_𝑘(𝑧), that is, 𝐻_𝑝𝑆_𝑝=𝐼_|𝐹.
Now, as [𝐻_𝑝]^𝑘𝑒_𝑛=0 for all 𝑘>𝑛≥𝑝 we deduce that any element of 𝐹 can be annihilated by a finite power 𝑘_𝑛 of 𝐻_𝑝 since as [∏^{𝑘𝑛+𝑛}_𝑗=𝑛𝜔_𝑗]⁻¹→0 when 𝑘_n→∞, we have𝑆^𝑘𝑛_𝑝𝑒_𝑛=^{𝑘𝑛+𝑛}_𝑗=𝑛𝜔_𝑗⁻¹𝑒_𝑘+𝑛⟶0in𝐵_𝑝. (2.1) Now, the hypercyclicity of 𝐻_𝑝 follows from the theorem of Bés et al. recalled above.

Lemma 2.7. Let 𝐻_𝑝=𝑧^𝑝(𝑑^𝑝+1/𝑑𝑧^𝑝+1) with domain 𝐷(𝐻_𝑝)={𝜙∈𝐵;𝐻_𝑝𝜙∈𝐵}⋂𝐵_𝑝, where 𝐻_𝑝𝑒_𝑛=𝜔_𝑛−1𝑒_𝑛−1, 𝑒_𝑛(𝑧)=(𝑧^𝑛/√𝑛!), and 𝜔_𝑛=√𝑛+1(𝑛!/(𝑛−𝑝)!) for 𝑛≥𝑝≥0. Then, there exist 𝑘>0 and 𝑔∈𝐷(𝐻^𝑘_𝑝) such that 𝐻^𝑘_𝑝𝑔(𝑧)=𝑔(𝑧).

Proof. Let 𝜆∈ℂ and 𝑔_𝜆(𝑧)=𝑒_𝑝(𝑧)+^∞_𝑛=𝑝+1𝜆^𝑛−𝑝𝜔_𝑝𝜔_𝑝+1⋯𝜔_𝑛−1𝑒_𝑛(𝑧), (2.2) then 𝑔_𝜆 is in the domain of 𝐻_𝑝 and it is an eigenvector for 𝐻_𝑝 corresponding to eigenvalue 𝜆, therefore it is a periodic point of 𝐻_𝑝. 𝜆 is a root of unity.
In fact, let 𝑟>0 and |𝜆|<𝑟, then as lim^𝑛−1_𝑗=𝑝𝜔_𝑗=∞when𝑛⟶∞, (2.3) there exist 𝑛₀>0 and 𝑞<1 such that 𝑟𝜔_𝑝𝜔_𝑝+1⋯𝜔_𝑛−1^(1/𝑛)≤𝑞for𝑛≥𝑛₀, (2.4) since for |𝜆|<𝑟, we have |𝜆|^{(𝑛−𝑝)}𝜔_𝑝𝜔_𝑝+1⋯𝜔_𝑛−1²≤𝑞^2𝑛;𝑛≥𝑛₀ (2.5) and 𝑔_𝜆 is in Bargmann space. Now as, 𝑔_𝜆,𝑒_𝑝𝑔=1,_𝜆,𝑒_𝑛+1=𝜆^{𝑛−𝑝+1}𝜔_𝑝𝜔_𝑝+1⋯𝜔_𝑛, (2.6) we get 𝑔_𝜆,𝑒_𝑛+1²=𝜆^{2(𝑛−𝑝+1)}𝜔_𝑝𝜔_𝑝+1⋯𝜔_𝑛²,|𝑔_𝜆,𝑒_𝑛+1|²𝜔_𝑛²=𝜆^{2(𝑛−𝑝+1)}𝜔_𝑝𝜔_𝑝+1⋯𝜔_𝑛−1²≤𝑞^2𝑛|𝜆|²for𝑛≥𝑛₀. (2.7) We deduce that^∞_𝑛=𝑝𝑔_𝜆,𝑒_𝑛+1²𝜔_𝑛²<∞, (2.8)
that is, 𝑔_𝜆∈𝐷(𝐻_𝑝).
Thus, we get𝐻_𝑝𝑔_𝜆=𝐻_𝑝𝑒_𝑝+^∞_𝑛=𝑝+1𝜆^𝑛−𝑝𝜔_𝑝𝜔_𝑝+1⋯𝜔_𝑛−1⋅𝜔_𝑛−1𝑒_𝑛−1=𝜆^∞_𝑚=𝑝𝜆^𝑚−𝑝𝜔_𝑝𝜔_𝑝+1⋯𝜔_𝑚−1𝑒_𝑚=𝜆𝑒_𝑝+^∞_𝑚=𝑝+1𝜆^𝑚−𝑝𝜔_𝑝𝜔_𝑝+1⋯𝜔_𝑚−1𝑒_𝑚=𝜆𝑔_𝜆. (2.9)
Therefore, 𝑔_𝜆 is the eigenvector corresponding to the eigenvalue 𝜆 and 𝑔_𝜆 is a periodic point of 𝐻_𝑝, where 𝜆 is a root of unity.

Lemma 2.8. The set of periodic points of 𝐻_𝑝 is dense in 𝐵_𝑝.

Proof. Let 𝐺=span𝑔_𝜆;𝜆isarootofunity. (2.10)𝐺 is dense in 𝐵_𝑝, otherwise there exists nonzero vector 𝑔∈𝐵_𝑝 which is orthogonal to 𝐺.
Let 𝑔(𝑧)=∑^∞_𝑛=𝑝𝑏_𝑛𝑒_𝑛(𝑧)suchthat𝑔,𝑔_𝜆=0,foreach𝑔_𝜆𝑓(𝜆)=∈𝐺,𝑔,𝑔_𝜆for|𝜆|<1,𝑓(𝜆)=0for|𝜆|=1. (2.11)
𝑓(𝜆) 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 𝑓(𝜆) vanishes for all |𝜆|≤1. We deduce that 𝑏_𝑛=0 for 𝑛≥𝑝, then 𝐺 is dense in 𝐵_𝑝.

Remark 2.9. (i) The Lemmas 2.5, 2.6, and 2.8 show the chaoticity of 𝐻_𝑝.
(ii) The Theorem 2.3 generalizes the result of [12] on the annihilation operator in Bargmann space.

Definition 2.10. Let 𝑇 be an unbounded linear operator on a separable infinite dimensional Banach 𝑋 whose domain 𝐷(𝑇) is dense in 𝑋, and let 𝑇^𝑛 be closed for all positive integers 𝑛.(a)A closed subspace 𝐸⊂𝑋 has hyperbolic structure if 𝐸=𝐸^𝑢⊕𝐸^𝑠, 𝑇𝐸^𝑢=𝐸^𝑢, and 𝑇𝐸^𝑠=𝐸^𝑠, where 𝐸^𝑢 (the unstable subspace) and 𝐸^𝑠 (the stable subspace) are closed. In addition, there exist constants 𝜏(0<𝜏<1) and 𝐶>0, such that:(i)For any Φ∈𝐸^𝑢,𝑘∈ℕ,𝐶𝜏^−𝑘||Φ||≤||𝑇^𝑘Φ|| (the vectors of 𝐸^𝑢 are exponentially expanded, we say they belong to the unstable subspace 𝐸^𝑢).(ii)For any Ψ∈𝐸^𝑠,𝑘∈ℕ,||𝑇^𝑘Ψ||≤𝐶𝜏^𝑘||Ψ|| (some vectors are contracted exponentially fast by the iterates of the operator 𝑇, we say they belong to the stable subspace 𝐸^𝑠).(b)If there exists a closed subspace 𝐸⊂𝑋 which has hyperbolic structure relative to 𝑇 and the set of periodic points of 𝑇 is dense in 𝐸, then 𝑇 is said to be a nonwandering operator relative to 𝐸 following the definition of Tian et al. [8].
Since 𝐻_𝑝 is chaotic operator on 𝐵_𝑝, so 𝐻_𝑝 has dense set of periodic points on 𝐵_𝑝, we only need to construct an hyperbolic structure associated to it in 𝐵_𝑝 to obtain:

Theorem 2.11. Let 𝐵 be the Bargmann space with orthonormal basis 𝑒_𝑛(𝑧)=𝑧^𝑛/√𝑛!.
Let 𝐻_𝑝=𝑧^𝑝(𝑑^𝑝+1/𝑑𝑧^𝑝+1)=𝐴^∗𝑝𝐴^𝑝+1 with domain 𝐷(𝐻_𝑝)={𝜙∈𝐵;𝐻_𝑝𝜙∈𝐵}⋂𝐵_𝑝, where 𝐵_𝑝={𝜙∈𝐵;(𝑑^𝑗/𝑑𝑧^𝑗)𝜙(0)=0,0≤𝑗≤𝑝}.
Then, 𝐻_𝑝 is a nonwandering operator.

Proof. We construct a closed invariant subspace 𝐸⊂𝐵_𝑝 such that 𝐸 has hyperbolic structure.
For 𝜆∈ℂ, the function defined by (2.2).
𝑔_𝜆(𝑧)=𝑒_𝑝(𝑧)+∑^∞_𝑛=𝑝+1(𝜆^𝑛−𝑝/𝜔_𝑝𝜔_𝑝+1⋯𝜔_𝑛−1)𝑒_𝑛(𝑧) is in the domain of 𝐻_𝑝 and is an eigenvector for 𝐻_𝑝 corresponding to the eigenvalue 𝜆.
Let 𝐸^𝑢=span{𝑔_𝜆;|𝜆|>1}, 𝐸^𝑠=span{𝑔_𝜆;0<|𝜆|<1}, and 𝐸=𝐸^𝑢⊕𝐸^𝑠, where ⊕ represents direct sum.
We will verify that 𝐸 has an hyperbolic structure.
For 𝜙∈𝐸^𝑢, there exists a sequence (𝑎_𝑖),𝑖=1,2,… such that𝜙(𝑧)=^∞_𝑖=1𝑎_𝑖𝑔_𝜆𝑖(𝑧)=^∞_𝑖=1𝑎_𝑖^∞_𝑛=𝑝𝜆_𝑖^𝑛−𝑝𝜔_𝑝𝜔_𝑝+1⋯𝜔_𝑛−1𝑒_𝑛(𝑧). (2.12) And for each positive integer 𝑚, we have𝐻_𝑝^𝑚𝜙=𝐻_𝑝^𝑚∞_𝑖=1𝑎_𝑖𝑔_𝜆𝑖=𝐻_𝑝^𝑚∞_𝑖=1𝑎_𝑖𝜆^𝑚_𝑖^∞_𝑛=𝑝𝜆_𝑖^𝑛−𝑝𝜔_𝑝𝜔_𝑝+1⋯𝜔_𝑛−1𝑒_𝑛≥𝜇^𝑚𝜙, (2.13)
where𝜇=min{|𝜆_𝑖|;|𝜆_𝑖|>1}.
Next, we will prove 𝐸^𝑢 is the invariant subspace of 𝐻_𝑝.
Let 𝜙∈𝐸^𝑢, then𝜙(𝑧)=^∞_𝑖=1𝑎_𝑖𝑔_𝜆𝑖(𝑧)=^∞_𝑖=1𝑎_𝑖^∞_𝑛=𝑝𝜆_𝑖^𝑛−𝑝𝜔_𝑝𝜔_𝑝+1⋯𝜔_𝑛−1𝑒_𝑛(𝑧)=𝐻_p^∞_𝑖=1𝑏_𝑖𝑔_𝜆i(𝑧), (2.14) where𝑏_𝑖=𝑎_𝑖/𝜆_𝑖then𝐸^𝑢⊂𝐻_𝑝𝐸^𝑢.
Now for 𝜓∈H_pE^u, then there exists 𝜙∈E^𝑢, such that 𝜓=𝐻_𝑝Φ=∑^∞_𝑖=1𝑐_𝑖𝑔_𝜆𝑖, where 𝑐_𝑖=𝜆_𝑖𝑎_𝑖. Therefore, 𝐻_𝑝𝐸^𝑢⊂𝐸^𝑢.
Similarly, let 𝐸^𝑠=span{𝑔_𝜆;0<|𝜆|<1}, we deduce that 𝐻_𝑝𝐸^𝑠=𝐸^𝑠 and if we chose 𝜏=1/𝜇, then we have 𝐻_𝑝^𝑚𝜙≥𝜙𝜏^𝑚. (2.15) Then, 𝐸 has hyperbolic structure and 𝐻_𝑝 is nonwandering operator relative to 𝐸.
Here, the linear space 𝐸^𝑠 is formed by the (spectral) subspace corresponding to the eigenvalues of 𝐻_𝑝 of modulus less than 1, while the unstable subspace 𝐸^𝑢 corresponds to those of modulus greater than 1.

As in [22], we can use the Gazeau-Klauder formalism to construct the coherent states of this operator 𝐻_𝑝 and investigate some properties of these coherent states (see [23]).

We conclude that main results of this work can be considered in [24] as an introduction to study of the operators 𝐻_𝑝,𝑚=𝑧^𝑝(𝜕^𝑝+𝑚/𝑑𝑧^𝑝+𝑚) with 𝑝=0,1,… and 𝑚=1,2,… particularly, to study the chaoticity of 𝐻_𝑝,𝑚+𝐻^∗_𝑝,𝑚.

Appendix

Let us recall below the essential spaces and operators used in above sections(i)𝐵={𝜙∶ℂ→ℂentire;∫_ℂ|𝜙(𝑧)|²𝑒^−|𝑧|2𝑑𝑥𝑑𝑦<∞},(ii)𝐵_𝑝={𝜙∈𝐵;(𝑑^𝑗/𝑑𝑧^𝑗)𝜙(0)=0,0≤𝑗≤𝑝},(iii)𝐻_𝑝𝑒_𝑛=𝜔_𝑛−1𝑒_𝑛−1 with 𝑒_𝑛(𝑧)=𝑧^𝑛/√𝑛! and 𝜔_𝑛=√𝑛+1(𝑛!/(𝑛−𝑝)!) for 𝑛≥𝑝≥0, (iv)𝐹={𝜙_𝑘(𝑧)=∑^𝑘_𝑛=𝑝𝑎_𝑛𝑒_𝑛(𝑧),𝑘=𝑝,𝑝+1,…},(v)𝑆_𝑝𝑒_𝑛=(1/𝜔_𝑛)𝑒_𝑛+1;𝑛≥𝑝≥0.

Then, we have the following.

Lemma A.1. For arbitrary 𝜙,𝜓∈𝐵_𝑝, there exists 𝜙_𝑘∈𝐹 such that 𝜙_𝑘→𝜙 and 𝐻^𝑘_𝑝𝜙_𝑘→𝜓.

Proof. As 𝐹 is dense in 𝐵_𝑝, then for arbitrary 𝜓∈𝐵_𝑝, there exists 𝜓_𝑘∈𝐹 such that 𝜓_𝑘→𝜓.
Let m a natural number, as 𝜔_𝑛=√𝑛+1(𝑛!/(𝑛−𝑝)!) for 𝑛≥𝑝≥0, then we get𝜔_𝑛𝜔_𝑛+1⋯𝜔_𝑛+𝑚=(𝑛+𝑚)!^𝑛+𝑚_𝑗=𝑛𝑗!𝑗−𝑝!for1𝑛≥𝑝≥0,𝜔_𝑛𝜔_𝑛+1⋯𝜔_𝑛+𝑚≤1(𝑛+𝑚)!for𝑛≥𝑝≥0. (A.1) Now, for arbitrary 𝜙∈𝐹 and ||𝜙||=1, 𝜙=∑^∞_𝑛=𝑝𝑎_𝑛𝑒_𝑛, we have 𝑆^𝑚_𝑝𝜙=^∞_𝑛=𝑝𝑎_𝑛1𝜔_𝑛𝜔_𝑛+1⋯𝜔_𝑛+𝑚𝑒_𝑛+𝑚,𝑆^𝑚_𝑝𝜙≤^∞_𝑛=𝑝𝑎_𝑛1()!.𝑛+𝑚 (A.2)
As for 𝑚≥3, we have (1/𝑚!)≤(1/2^𝑚), then 𝑆^𝑚_𝑝𝜙≤1√2^𝑚+1√2^𝑚+11+⋯≡2−√21√2^𝑛⟶0(𝑚⟶∞), (A.3) hence 𝑆^𝑚_𝑝 tends pointwise to zero on 𝐹.
By choosing diagonal element of {𝑆^𝑚_𝑝𝜓_𝑘}, we get𝑆^𝑘_𝑝𝜓_𝑘⟶0when𝑘⟶∞. (A.4) As 𝐻_𝑝𝑆_𝑝=𝐼 on 𝐹, then we can write 𝜓_𝑘=𝐻_𝑝𝑆_𝑝𝜓_𝑘, that is, 𝐻_𝑝𝑆^𝑘_𝑝𝜓_𝑘⟶𝜓when𝑘⟶∞. (A.5)
Now as 𝐾𝑒𝑟𝐻^𝑚_𝑝=span{𝑒_𝑗,𝑝≤𝑗≤𝑝+𝑚−1}, where 𝐾𝑒𝑟𝐻^𝑚_𝑝, is the kernel space of 𝐻^𝑚_𝑝 then ∏^∞_𝑚=𝑝𝐾𝑒𝑟𝐻^𝑚_𝑝, is dense in 𝐵_𝑝 and for arbitrary Φ∈⋃∏^∞_𝑚=𝑝𝐾𝑒𝑟𝐻^𝑚_𝑝, there exists 𝑚 such that 𝜙∈𝐾𝑒𝑟𝐻^𝑚_𝑝. Therefore, 𝐻^𝑚_𝑝 tends pointwise to zero on a dense subset of 𝐵_𝑝.
For arbitrary 𝜙∈𝐵_𝑝, there exists 𝑓_𝑘∈∏^∞_𝑚=𝑝𝐾𝑒𝑟𝐻^𝑚_𝑝 such that 𝑓_𝑘→𝜙, therefore𝐻^𝑚_𝑝𝑓_𝑘⟶0(𝑚⟶∞). (A.6) Particularly, 𝐻^𝑘_𝑝𝑓_𝑘⟶0(𝑘⟶∞). (A.7) Let 𝜙_𝑘=𝑓_𝑘+𝑆^𝑘_𝑝𝜓_𝑘. (A.8) Then, 𝐻^𝑘_𝑝𝜙_𝑘→𝜓(𝑘→∞).

Lemma A.2. Let 𝜙∈𝐺=⋂^∞_𝑗=0⋃^∞_𝑚=0𝐻^𝑚_𝑝𝐷_𝑗, where 𝐷_𝑗, is an enumeration of open ball in 𝐵_𝑝 with centers in a countable dense subset of 𝐵_𝑝, then {𝜙,𝐻_𝑝𝜙,…,𝐻^𝑚_𝑝Φ,…} is dense in 𝐵_𝑝.

Proof. The above lemma imply for arbitrary 𝜙∈𝐷_𝑗 and 𝜓∈𝐵_𝑝 that there exists 𝜙_𝑘∈𝐷_𝑗 such that 𝜙_𝑘→𝜙, and 𝐻^𝑘_𝑝𝜙_𝑘→𝜓, hence ⋃^∞_𝑚=0𝐻^𝑚_𝑝𝐷_𝑗 is dense in 𝐵_𝑝 and Baire category theorem implies 𝐺 is dense in 𝐵_𝑝. Hence, 𝐻_𝑝 is topologically transitive.