On chaoticity of the sum of chaotic shifts with their adjoints in Hilbert space and applications to some chaotic weighted shifts acting on some Fock-Bargmann spaces
aa r X i v : . [ m a t h . F A ] N ov On chaoticity of the sum of chaotic shifts with their adjointsin Hilbert space and applicationsto some chaotic weighted shifts actingon some Fock-Bargmann spaces
Abdelkader Intissar
Equipe d’Analyse spectrale,Facult´e des Sciences et Techniques
Universit´e de Cort´e, 20250 Cort´e, France
T´el: 00 33 (0) 4 95 45 00 33-Fax: 00 33 (0) 4 95 45 00 [email protected] & Le Prador, 129, rue du Commandant Rolland, 13008 Marseille, France
Abstract
This article is intended to outline some the recent work by the authoron the chaoticity of some specific bakward shift unbounded operatorsrealized as differential operators acting on some Fock- Bargmann spacesand give sufficient conditions on a linear unbounded densely definedchaotic shift operator T acting on a Hilbert space for the operator T + T ∗ to be chaotic where T ∗ is its adjoint.. Mathematics Subject Classification:
Keywords:
Chaotic operators; weighted shift unbounded operators; analyticfunctions; Gamma function;quantization of the complex unit disk; 2D-Zernikepolynomials on the unit disk; Fock- Bargmann spaces;non-compact lattice of(Γ , χ )-theta Fock-Bargmann space ; reproducing kernels; Gelfond-Leontiev op-erators;Spectral analysis. n chaoticity of the sum of weighted shift with its adjoint 2013 September 16
A continuous operator T on a Banach space X is said to be hypercyclic if thefollowing condition is met:There exists an element φ ∈ X that its orbit Orb ( T , φ ) = { φ, T φ, T φ, ..... } is dense in X and is said to be chaotic in the sense of Devaney [2,15] if thefollowing conditions is met:1) T is hypercyclic.2) The set { φ ∈ X ; ∃ n ∈ IN such that T n φ = φ } of periodic points of opera-tor T is dense in X .It is well known that linear operators in finite-dimensional linear spaces can’tbe chaotic but the nonlinear operator may be. Only in infinite-dimensionallinear spaces can linear operators have chaotic properties.These last proper-ties are based on the phenomenon of hypercyclicity or the phenomen of non-wandercity.The study of the phenomenon of hypercyclicity originates in the papers byBirkoff [7] and Maclane [27] that show, respectively, that the operators oftranslation and differentiation, acting on the space of entire functions are hy-percyclic.The theories of hypercyclic operators and chaotic operators have been inten-sively developed for bounded linear operator, we refer to [13,14] and referencestherein and for a bounded operator, Ansari asserts in [1] that powers of a hy-percyclic bounded operator are also hypercyclicFor an unbounded operator, Salas exhibit in [30] an unbounded hyper-cyclic operator whose square is not hypercyclic. The result of Salas show thatone must be careful in the formal manipulation of operators with restricteddomains. For such operators it is often more convenient to work with vectorsrather than with operators themselves.Now, let T be an unbounded operator on a separable infinite dimensional Ba-nach space X .We define the following sets: D ( T ) = { φ ∈ X ; T φ ∈ X } (1 . D ( T ∞ ) = T ∞ n =0 D ( T n ) (1 . [6] by B´es etal as follows: Definition 1.1.
A linear unbounded densely defined operator ( T , D ( T )) on A. Intissarn chaoticity of the sum of weighted shift with its adjoint 2013 September 16 a Banach space X is called chaotic if the following conditions are met:1) T n is closed for all positive integers n ..2) there exists an element φ ∈ D ( T ∞ ) whose orbit Orb ( T , φ ) = { φ, T φ, T φ, ..... } is dense in X
3) the set { φ ∈ X ; ∃ m ∈ IN such that T m φ = φ } of periodic points of opera-tor T is dense in X . Recently these theories are begin developed on some concrete examples ofunbounded linear operators, see [5,8,16] . In [16] it has been shown that theoperators H p = z p d p +1 dz p +1 ; p = 0 , , ..... are chaotic in the sense of Definition 1.1on the classic Bargmann space [3] of entire functions with e −| z | measure.In [17] we have considered generalized Bargmann spaces (the spaces ofentire functions with e −| z | β measure; β >
0) and we have proved that the op-erators H p = z p D p +1 ; p = 0 , , ..... in these spaces are chaotic where D is theadjoint operator of the operator of multiplication by the independent variable z on these spaces. D belongs to class Gelfond-Leontiev operators of generalizeddifferentiation [10] In [18] we have considered non-compact lattice of (Γ , χ )-theta Fock-Bargmannspaces and we have proved that the operators H p = e ipz D p +1 ; p = 0 , , ..... inthese spaces are chaotic where D is the adjoint operator of the operator ofmultiplication by the function M ( z ) = e ipz on these spaces.In the present work, we give sufficient conditions on a linear unboundeddensely defined chaotic shift operator T acting on a Hilbert space such that T + T ∗ is chaotic where T ∗ is its adjoint and we apply this sufficient conditionsto above operators.In appendix, we consider Fock-Bargmann space on the unit disk quantizedwhere we will consider the annihilation operator A associated its orthonormalbasis and we prove that the operators H p = A ∗ p A p +1 ; p = 0 , , ..... are chaoticwhere A ∗ is the adjoint operator of the annihilation operator A .This paper is organized as follows :In section 2 we recall some sufficient conditions on hypercyclicity of boundedoperators given by Godefroy-Shapiro’s lemma [12] or on hyperccylicity of un-bounded operators given by B` e s-Chan-Seubert theorem [6] .We recall also some elementary properties of classic Bargamann space, thegeneralized Bargmann space, the (Γ , χ )-theta Fock-Bargmann space, Fock-Bargmann space on the unit disk quantized and the action of H p ; p ∈ N onthese spaces.In section 3 we give sufficient conditions on a linear unbounded densely defined3 A. Intissarn chaoticity of the sum of weighted shift with its adjoint 2013 September 16 chaotic shift operator T acting on a Hilbert space such that T + T ∗ is chaoticwhere T ∗ is its adjoint with application to H p ; p ∈ N defined on above spaces.In appendix we consider the annihilation operator A = ddz acting on orthonor-mal basis of the Fock-Bargmann space on the unit disk quantized. We provethat the operators H p = A ∗ p A p +1 ; p = 0 , , ..... acting on this space are chaoticwhere A ∗ = z ddz + 2 νz ; ν > A .As these operators are unilateral weighted backward shifts with an explicitweight, we use the results of B` e s et al to proof the chaoticity of H p on Fock-Bargmann space associated to Poincar´e disk (we can also use the results ofBermudez et al [5] to proof the chaoticity of our operators H p ). H p of order p on associ-ated Fock-Bargmann spaces Before to recall the Fock-Bargmann spaces those the operators H p with do-main D ( H p ) acting, we begin by to recall that an unbounded operator T ishypercyclic if there is a vector φ in the domain of T such that for every integer m > T m φ is in the domain of T and the orbit { φ, T φ, T φ, T φ, ... } is dense in X .We recall also the Godefroy-Shapiro’s lemma [12] and B` e s-Chan-Seubert’stheorem [6] that we will used in section 3 and in appendixA) hypercyclicity criterion of Godefroy-Shapiro and of B` e s-Chan-Seubert Lemma 2.1. (Godefroy-Shapiro ([12]).Let X be a separable Fr´echet space and T is bounded operator on X and Y , Y are two dense subsets of X and S : Y → Y such that :(1) TS φ = φ, ∀ φ ∈ Y (2) lim S m φ = 0 , ∀ φ ∈ Y as m → + ∞ (3) lim T m φ = 0 , ∀ φ ∈ Y as m → + ∞ then T is hypercyclic operator. then hypercyclic vectors can be constructed for an unbounded operator T , under a sufficient condition analogous to the above hypercyclicity criterionwitch is given by B ` es − Chan − Seubert ’s theorem4
A. Intissarn chaoticity of the sum of weighted shift with its adjoint 2013 September 16
Theorem 2.2. (B ` e s-Chan-Seubert [6] , p.258 )Let X be a separable infinite dimensional Banach and let T be a densely definedlinear operator on X . Then T is hypercyclic if(i) T m is closed operator for all positive integers m .(ii) There exist a dense subset Y of the domain D ( T ) of T and a (possiblynonlinear and discontinuous) mapping S : Y −→ Y so that TS = I | Y ( I | Y isidentity on Y ) and T n , S n −→ pointwise on Y as n −→ ∞ . B) the classic Bargmann space [3] is defined by: B = { φ : IC −→ IC entire ; Z IC | φ ( z ) | e −| z | dxdy < ∞} (2 . z = x + iy . B is a Hilbert space with an inner product < φ, ψ > = Z IC φ ( z ) ψ ( z ) e −| z | dxdy (2 . || . || .The functions e n ( z ) = z n √ n ! ; n = 0 , , , .... form a complete orthonormal setin B The operator of multiplication by the independent variable z on B is de-fined by : A ∗ φ ( z ) = zφ ( z ) with domain D ( A ∗ ) = { φ ∈ B ; zφ ∈ B } (2 . A ∗ acts on e n ( z ) as following: A ∗ e n ( z ) = √ n + 1 e n +1 ( z ) = ω n e n +1 ( z ) with ω n = √ n + 1 (2 . A φ ( z ) = ddz φ ( z ) with domain D ( A ) = { φ ∈ B ; ddz φ ∈ B } (2 . A e ( z ) = 0 and A e n ( z ) = √ ne n − ( z ) = ω n − e n − ( z ) , n ≥ . H p acting on B as following5 A. Intissarn chaoticity of the sum of weighted shift with its adjoint 2013 September 16 H p = A ∗ p A p +1 with domain D ( H p ) = { φ ∈ B ; H p φ ∈ B } (2 . H ∗ p e n ( z ) = A ∗ p +1 A p e n ( z ) = p ( n + 1) Q pj =1 ( n − j ) e n +1 ( z ) for n ≥ p ≥ . H ∗ p is weighted shift with weight ω n,p = ω n Q pj =1 ω n − j for n ≥ p ≥ . F β = { φ : IC −→ IC entire ; Z IC | φ ( z ) | e −| z | β dµ ( z ) < ∞} (2 . β > dµ ( z ) = β π Γ( β ) dxdy and z = x + iy .Note that F coincides with the classic Bargmann space. F β is a Hilbert space with an inner product < φ, ψ > = β π Γ( β ) Z IC φ ( z ) ψ ( z ) e −| z | β dxdy (2 . || . || .Let m = 0, m n = Γ( β ( n +1))Γ( nβ ) n = 1 , , ... and [ m n ]! = m .m ......m n then itmay be shown that the functions e ( z ) = 1 and e n ( z ) = z n √ [ m n ]! ; n = 1 , , .... (2 . F β .Define the principal vectors e λ ∈ F β (for every λ ∈ IC ) as complex valuedfunctions e λ ( z ) = e ( z, λ )= 1 + ∞ X n =1 e n ( z ) e n ( λ ) of λ and z in IC If φ ( z ) = ∞ X n a n e n ( z ) then < φ, e λ > = φ ( λ ) ( the reproducing property)6 A. Intissarn chaoticity of the sum of weighted shift with its adjoint 2013 September 16 because Z IC ∞ X n a n e n ( z )(1 + ∞ X n =1 e n ( z ) e n ( λ )) e −| z | β dµ ( z ) = a + ∞ X n =1 a n e n ( λ ) || e n || = φ ( λ )or, in other words φ ( z ) = Z IC φ ( λ ) e λ ( z ) e −| λ | β dµ ( λ ) for all φ ∈ F β (2 . e λ ( z ) is called a reproducing kernel for F β Note that the reproducing kernel e λ ( z ) is uniquely determined by theHilbert space F β and the evaluation linear functional φ ∈ F β → φ ( z ) ∈ IC is a bounded linear functional on F β .So applying (2.13) to the function e z at λ ; we get e z ( λ ) = < e z ; e λ > for z ; λ ∈ IC and by the above relations, for z ∈ IC we obtain || e z || = √ < e z , e z > = p e ( z, z ). (2 . [24] where Irac-Astaud and Rideau have constructed an deformed har-monic algebra (DHOA) on F β and in [25] where Knirsch and Schneider haveinvesigated the continuity and Schattenvon Neumann p -class membership ofHankel operators with anti-holomorphic symbols on these spaces with β ∈ N .Note that the generalized Bargmann spaces F β are different from the general-ized Bargmann spaces E m m = 0 , , .... defined in [19] . It would be interestingto characterize the orthogonal space of F β in L ( C , e −| z | β dµ ( z )) for β = 2.On the generalized Bargmann representation F β , we denote now the oper-ator of multiplication by the independent variable z on F β by : M φ ( z ) = zφ ( z ) with domain ID ( M ) = { φ ∈ F β ; zφ ∈ F β } (2 . M acts on e n ( z ) as following: M e n ( z ) = q Γ( β ( n +2)) q Γ( β ( n +1)) e n +1 ( z ) (2 . D e n ( z ) = q Γ( β ( n +1)) q Γ( β n ) e n − ( z ) (2 . φ ( z ) = ∞ X n =0 a n z n we have D D φ ( z ) = 1 z ∞ X n =0 a n m n z n where7 A. Intissarn chaoticity of the sum of weighted shift with its adjoint 2013 September 16 m n = Γ( β ( n +1))Γ( β n ) with domain: D ( D ) = { φ ∈ F β ; D φ ∈ F β } (2 . β = 2 the generalized differentiation operator D is: D φ ( z ) = ddz φ ( z ) (2 . H p acting on F β as following H p = M p D p +1 with domain D ( H p ) = { φ ∈ F β ; H p φ ∈ F β } (2 . H ∗ p e n ( z ) = M p +1 D p e n ( z ) = √ m n +1 Q pj =1 [ m n − j +1 ] e n +1 ( z ) for n ≥ p ≥ H ∗ p is weighted shift with weight ω n,p = √ m n +1 Q pj =1 [ m n − j +1 ] for n = 1 , ..... and as we have denoted [ m n ]! = m .m .......m n then ω n,p = √ m n +1 [ m n ]![ m n − p ]! for n ≥ p ≥ Remark 2.3. (i) If β = 2 and p = 0 then the operator H = D is particularcase of Gelfond-Leontiev operator of generalized differentiation [10] on F β andcoincides with the usual differentiation on F .(ii) For β = 2 , It is known in [16] that :(a) the operator H p with its domain D ( H p ) is an operator chaotic on the classicBargmann space.(b) H φ λ ( z ) = λφ λ ( z ) ∀ λ ∈ IC , where φ λ ( z ) = ∞ X n =0 λ n √ n ! e n ( z ) and || φ λ || = e | λ | (c) The function e −| λ | φ λ ( z ) is called a coherent normalized quantum optics(see [26] )(d) For p = 1 , it is known that H + H ∗ is a not selfadjoint operator andchaotic on the classic Bargmann space [8] . This operator play an essentialrole in Reggeon field theory (see [20] , [21] and [22] ) D) Let z = x + iy , ν >
0, Γ = Z ω the discrete subgroup of the additivegroup ( IC, +) where ω ∈ C − { } and χ be a given map χ : Z ω → U (1) = { λ ∈ IC ; | λ | = 1 } such that χ ( γ ) = e iπαm for γ = mω ∈ Z ω , where α is a fixedreal number. Γ is non-compact lattice of the rank one and the triplet ( ν, Γ , χ )satisfies a Riemann-Dirac quantization type condition.8 A. Intissarn chaoticity of the sum of weighted shift with its adjoint 2013 September 16 In [11] Ghanmi and Intissar have considered the space L ,ν Γ ,χ ( IC ) of all measur-able functions on IC that are integrable on V (Γ) with respect e − ν | z | dxdy andsatisfying the function equation : φ ( z + γ ) = χ ( γ ) e νzγ + ν | γ | φ ( z ) (2 . z ∈ C and every γ ∈ Γ,where V (Γ) is any given fundamentaldomain of the lattice Γ witch is the strip S = [0 , × R . L ,ν Γ ,χ ( IC ) is Hilbert space with the inner scalar product : < φ , φ > Γ = Z IC/ Γ φ ( z ) φ ( z ) e − ν | z | dxdy = Z V (Γ) φ ( z ) φ ( z ) e − ν | z | dxdy (2 . || φ || Γ = sZ V (Γ) | φ ( z ) | e − ν | z | dxdy (2 . ω = 1, we write the function equation (2 .
1) in the form: φ ( z + m ) = e iπαm e ν ( zγ + m ) m φ ( z ) (2 . L ,ν Γ ,χ ( IC ) by L ,ν Γ ,α ( IC )The (Γ , χ )-theta Fock-Bargmann space F ,ν Γ ,α ( IC ) is defined now as a sub-space of the space O ( IC ) of holomorphic functions on IC , given by F ,ν Γ ,α ( IC ) = O ( IC ) ∩ L ,ν Γ ,α ( IC ) = { φ ∈ O ( IC ); < φ, φ > Γ < ∞} (2 . e α,νn ( z ) = ( 2 νπ ) / e ν z e − π ν ( n + α ) +2 iπ ( n + α ) z ; n ∈ Z (2 . [11] , it is showed that the sequence e α,νn ( z ) n ∈ Z is orthonormal basis of F ,ν Γ ,α ( IC ) and a function φ ( z ) = X n ∈ Z a n e α,νn ( z )belongs to F ,ν Γ ,α ( IC ) if and only if X n ∈ Z | a n | < + ∞ We consider now the operator of multiplication M by the function M ( z ) = e iπz on the linear subspace F α of F ,ν Γ ,α ( IC ) generatedd by the orthogonal basis { e α,νn ( z ); n ∈ N } (here we formallly take e α,ν − ( z ) = 0) and D its adjoint.9 A. Intissarn chaoticity of the sum of weighted shift with its adjoint 2013 September 16
The operator M acts on e α,νn ( z ); n ∈ N as following: M e α,νn ( z ) = ω n e α,νn +1 ( z ) (2 . ω n = c α,ν e πν n and c α,ν = e πν +2 α can be identified with M ( a n ) n ∈ N = ( ω n a n +1 ) n ∈ N (2 . D e α,νn ( z ) = ω n − e α,νn − ( z ) (2 . D ( a n ) n ∈ N = ( ω n − a n − ) n ∈ N ; ω − = 0 (2 . Remark 2.4.
Let p ∈ N then D p +1 e α,νp ( z ) = 0 . We define a family of unilateral weighted shifts H p acting on F α as following H p = M p D p +1 with domain D ( H p ) = { φ ∈ F α ; H p φ ∈ F α } (2 . H p e α,νn ( z ) = M p D p +1 e α,νn ( z ) = ω n − [ p Y j =1 ω n − − j ] e α,νn − ( z ) (2 . H ∗ p e α,νn ( z ) = M p +1 D p e α,νn ( z ) = ω n [ p Y j =1 ω n − j ] e α,νn − ( z ) (2 . µ := 2 πν and c α := c α,ν = e ( µ +4 α ) and ω n,p := ω n [ p Y j =1 ω n − j ] (2 . A. Intissarn chaoticity of the sum of weighted shift with its adjoint 2013 September 16 ω n,p = ( c α ) p +1 e (2 p +1) µn (2 . H p e α,νn ( z ) = ω n − ,p e α,νn − ( z ) (2 . H p e α,νn ( z ) = 0 for p ≤ n and H ∗ p e α,νn ( z ) = ω n,p e α,νn +1 ( z ) (2 . D = { z ∈ C ; | z | < } , is involved as a fundamental model or at least is used as apedagogical toy (see for example Elwassouli et al in [9] ). It is a model of phasespace for the motion of a material particle on a one sheeted two-dimensionalhyperboloid viewed as a (1+1)-dimensional space-time with negative constantcurvature, namely, the two dimensional anti de Sitter space-time.The unit disk equipped with a K¨ a hlerian potential, K D ( z, z ) = π (1 − | z | ) ,has the structure of a two-dimensional K¨ a hlerian. Any K¨ a hlerian manifold issymplectic and so can be given a sense of phase space for some mechanicalsystem.Now let ν > be a real parameter and let us equip the unit disk D = { z ∈ C ; | z | < } with a measure dλ ν ( z ) = ν − π dxdy (1 −| z | ) .For ν >
1, we consider the Hilbert space L ν ( D , dµ ν ( z )) of all functions φ on D that are square integrable with respect to dµ ν ( z ) = (1 − | z | ) ν − dxdy (2 . FB ν = O ( D ) ∩ L ν ( D , dµ ν ( z )) (2 . O ( D ) is the space of all analytic functions φ ( z ) on D .Let z = re iθ with 0 < r < θ ∈ [0 , π ], with respect to the scalarproduct defined on the holomorphic polynomials by: < z n , z m > = Z D z n z m (1 − | z | ) ν − dxdy ; n ∈ N , m ∈ N (2 . A. Intissarn chaoticity of the sum of weighted shift with its adjoint 2013 September 16 Z D z n z m (1 − | z | ) ν − dxdy = Z π Z r n + m +1 e i ( n − m ) θ drdθ = 2 π Z t n (1 − t ) ν − dt = π ν − ν )Γ(2 ν + n ) Γ( n + 1) if m = n if m = n (2 . A = ddz is A ∗ = z ddz + 2 νz (2 . FB ν the scalar product < φ, ψ > D = 2 ν − π Z D φ ( z ) ψ ( z ) dµ ν ( z ) (2 . P n ( z ) = r (2 ν ) n n ! z n ; n ∈ N is an orthonormal basis of FB ν . (2 . ν ) n = Γ(2 ν + n )Γ(2 ν ) is the Pochhammer symbol.As on FB ν the adjoint operator of A = ddz is the differential operator A ∗ = z ddz + 2 νz then they act on the orthonormal basis P n ( z ) as following A ∗ P n ( z ) = ω n P n +1 ( z ) (2 . ω n = p ( n + 1)(2 ν + n ).and AP n ( z ) = ω n − P n − ( z ) where AP ( z ) = 0 (2 . H p acting on12 A. Intissarn chaoticity of the sum of weighted shift with its adjoint 2013 September 16 FB ν = O ( D ) ∩ L ν ( D , dµ ν ( z )); ν > H p = A ∗ p A p +1 ; p ∈ N be the linear unbounded densely defined shift opera-tor acting on the FB ν with domain D ( H p ) = { φ ∈ FB ν ; H p φ ∈ FB ν } whose its adjoint is defined by : H ∗ p P n = ω n,p P n +1 (2 . ω n,p = ω n p Y j =1 ω n − j ; n ≥ p ≥ . ω n,p = p ( n + 1)(2 ν + n ) p Y j =1 ( n − j + 1)(2 ν + n − j ) (2 . Remark 2.5.
For the above spaces we will consider only(i) the weights ω n,p = √ n + 1 p Y j =1 ( n − j + 1) for n ≥ p ≥ associated to shifts acting on classic Bargmann space by noting that in this case ω n,p ∼ n p + (2 . (ii) the weights ω n,p = √ m n +1 [ m n ]![ m n − p ]! for n ≥ p ≥ where m = 0 , [ m n ]! = m .m .......m n and m n = Γ( β ( n + 1))Γ( β n ) associated to shifts acting on generalized Bargmann space by noting that inthis case ω n,p ∼ n p +1 β (2 . (iii) the weights ω n,p = ( c α ) p +1 e (2 p +1) µn for n ≥ p ≥ . A. Intissarn chaoticity of the sum of weighted shift with its adjoint 2013 September 16 where µ := 2 πν and c α := c α,ν = e ( µ +4 α ) associated to shifts acting on theta- Fock-Bargmann space.(iv) the weights ω n,p = p ( n + 1)(2 ν + n ) p Y j =1 ( n − j + 1)(2 ν + n − j ) for n ≥ p ≥ associated to shifts acting on Fock-Bargmann space on Poincar´e disk bynoting that in this case ω n,p ∼ n p +1 (2 . In this section we give sufficient conditions on a linear unbounded densely de-fined chaotic shift operator T acting on a Hilbert space such that T + T ∗ ischaotic where T ∗ is its adjoint. Theorem 3.1.
Let a linear unbounded densely defined chaotic shift opera-tor ( T , D ( T )) on a Hilbert space E = { φ ; φ = ∞ X n =1 a n e n } such that its adjoint isdefined by: T ∗ e n = ω n e n +1 (3 . where { e n } is an orthonormal basis of E and ω n is positive weight associ-ated to T We assume that(Assumption
Hyp ) ∞ X n =1 ω n < ∞ (3 . (Assumption Hyp ) ω n − ω n +1 ≤ ω n (3 . A. Intissarn chaoticity of the sum of weighted shift with its adjoint 2013 September 16 (Assumption
Hyp ) there exist α > , β > , a > , and a sequence γ n that:(1) ω n γ n γ n +1 ≥ n α (3 . (2) ω n − γ n +1 ω n γ n − = 1 − an + O ( 1 n β ) (3 . and(3) ∞ X k =1 γ n < ∞ (3 . Then for λ ∈ C the following recurrence sequence ( ∗ ) u ( λ ) = 1 u ( λ ) = λω ω n − u n − ( λ ) + ω n u n +1 ( λ ) = λu n ( λ ) (3 . (i) is solvable for all λ ∈ C .(ii) ∞ X n =1 | u n ( λ ) | < ∞ for all λ ∈ C .(iii) the spectrum of T + T ∗ is the all complex plane C .(iv) ( T + T ∗ ) m is closed ∀ m ∈ N .(v) T + T ∗ is hypercyclic operator.(vi) T + T ∗ is chaotic operator. Proof of theorem (i) By using the Yu. Berzanskii’theory on the difference operators in [4] inparticular the theorem 1 . V II and the above asymptions (3.2) and (3.3)we deduce that the sequence u n ( λ ) is always solvable and is a polynomial ofdegree n − T .(ii) Let M > | u n ( λ ) |≤ Mγ n and | u n − ( λ ) |≤ Mγ n − .15 A. Intissarn chaoticity of the sum of weighted shift with its adjoint 2013 September 16 As ω n − u n − ( λ ) + ω n u n +1 ( λ ) = λu n ( λ ) Then u n +1 ( λ ) = λω n u n ( λ ) − ω n − ω n u n − ( λ ) then | u n +1 ( λ ) |≤ | λ | ω n | u n ( λ ) | + ω n − ω n | u n − ( λ ) |≤ M [ | λ | ω n γ n + ω n − ω n γ n − ] ≤ Mγ n +1 [ | λ | ω n γ n +1 γ n + ω n − ω n γ n +1 γ n − ]From (3.4) and (3.5) we get | u n +1 ( λ ) |≤ Mγ n +1 [ | λ | n α + 1 − an + O ( 1 n β )] | u n +1 ( λ ) |≤ Mγ n +1 [1 − an + | λ | n α + O ( 1 n β )] ≤ Mγ n +1 and from (3.6) we deduce that ∞ X n =1 | u n ( λ ) | < ∞ for all λ ∈ C (iii) Let u n ( λ ) the sequence defined by (3.7) and φ λ = ∞ X n =1 u n ( λ ) e n then( T + T ∗ ) φ λ = λφ λ .As ∞ X n =1 | u n ( λ ) | < ∞ for all λ ∈ C then the spectrum of T + T ∗ is C (iv) As T is chaotic we have T m , T ∗ m and T k T ∗ j are closed operators ∀ ( m, k, j ) ∈ N then ( T + T ∗ ) m is closed.(v) We verify now that the operator T + T ∗ on E satisfies the hypercyclicitycriterion, as quoted above.Let Ω = { λ ∈ C ; | λ | > } , Ω = { λ ∈ C ; | λ | < } , Ω = { λ ∈ C ; | λ | = 1 } and 16 A. Intissarn chaoticity of the sum of weighted shift with its adjoint 2013 September 16 F the space spanned by { φ λ = ∞ X n =1 u n ( λ ) e n ; λ ∈ Ω } F the space spanned by { φ λ = ∞ X n =1 u n ( λ ) e n ; λ ∈ Ω } F the space spanned by { φ λ = ∞ X n =1 u n ( λ ) e n ; λ ∈ Ω } each F j j = 1 , , E because if φ ∈ F ⊥ j then the function g ( λ ) = < φ, φ λ > is entire on C and equal to zero on F j witch has accumula-tion points, then g ( λ ) = 0 on C and also φ = 0.Let S be the linear mapping on F determined by: S φ λ = S ( N X n =1 a n φ λ n ) = N X n =1 a n δ n φ λ n ; λ n ∈ ω and a n ∈ C then ( T + T ∗ ) n → F , and TS = I and S n → F . So with the property (iv) we deduce that T + T ∗ is hypercyclic on E byusing the lemme of Godefroy-Shapiro or the theorem of B` e et al.(vi) For see that F is subset of periodic points of T + T ∗ , we take a m ∈ C , n m , k m ∈ Z , δ m = e iπ nmkm and φ = N X m =1 a m φ δ m . Then we observe that for l = Q Nm =1 k m we have ( T + T ∗ ) l φ = φ . Theorem 3.2.
Let B = { φ : C → C entire ; Z C | φ ( z ) | e −| z | dxdy < ∞} the classic Bargmann space and H p = z p d p +1 dz p +1 ; p ∈ N the linear unboundeddensely defined shift operator acting on the B p space; p = 0 , , ... B p = { φ ∈ B ; d j dz j φ (0) = 0; 0 ≤ j ≤ p } with domain D ( H p ) = { φ ∈ B ; H p φ ∈ B } ∩ B p whose its adjoint is defined by : H ∗ p e n = ω n,p e n +1 ((3 . A. Intissarn chaoticity of the sum of weighted shift with its adjoint 2013 September 16 where ω n,p = √ n + 1 f or p = 0 √ n + 1 p − Y j =0 ( n − j ) f or p ≥ . and { e n ( z ) = z n √ n ! ; n = p, p + 1 , .... } is an orthonormal basis of B p Then we have(i) for all p ≥ , H p is chaotic.(ii) for all p ≥ , H p + H ∗ p is chaotic. Proof (i) In [16] , the author showed that H p acting on B p is chaotic in the senseof Devaney for all p ∈ N .(ii) We will be concerned with the chaoticity of H p + H ∗ p for p ≥ .
2) of the above theorem is verified because ω n,p ∼ n p +1 / and as p ≥ ∞ X n =1 ω n,p < ∞ .-to verify the assumption (3 .
3) of the above theorem, we write ω n,p in thefollowing form ω n,p = √ n + 1 n ( n − p + 1) A n,p where A n,p = p − Y j =1 ( n − j )then ω n − ,p = √ n ( n − p + 1)( n − p ) A n,p A. Intissarn chaoticity of the sum of weighted shift with its adjoint 2013 September 16 and ω n +1 ,p = √ n + 1 n ( n + 1) A n,p and as n ≥ p we deduce that- ω n − ,p ω n +1 ,p ≤ ω n − ,p for all n ≥ p - ω n − ,p ω n,p = √ n ( n − p ) √ n + 1 n We choose choose γ n = √ nLogn (3 . | λ | ω n,p γ n +1 γ n = | λ | ω n,p √ n + 1 Log ( n + 1) √ nLogn ∼ n p +1 / - ω n − ,p ω n,p γ n +1 γ n − = n − p √ n √ n − Log ( n + 1) Log ( n − − pn )(1 − n ) − Log ( n + 1) Log ( n − − pn )(1 − n ) − ≤ − p − n + O ( 1 n )and Log ( n + 1) Log ( n −
1) =
Log ( n − Log ( n −
1) =
Log ( n −
1) +
Log (1 + n − ) Log ( n − C > Log ( n + 1) Log ( n − ≤ C ( n − Log ( n − a > β > | λ | ω n,p γ n +1 γ n + ω n − ,p ω n,p γ n +1 γ n − ≤ − an + 0( 1 n β ) ≤ n enough large.this implies that H p + H ∗ p is chaotic. Remark 3.3. (i) The operator µz p d p dz p + iλz p ( d p dz p + z p ) d p dz p with µ > , λ ∈ R and i = √− is not chaotic on B p , in fact it is an operator with com-pact resolvent. A. Intissarn chaoticity of the sum of weighted shift with its adjoint 2013 September 16
For p = 1 , the Reggeon field theory is governed by this non-self adjoint operatorsee [21] and references therein .(ii) We have assumed the assumptions Hyp1 and Hyp2 in theorem 3.1 forthe solvability of the sequence defined by (3.7): ( ∗ ) u ( λ ) = 1 u ( λ ) = λω ω n − u n − ( λ ) + ω n u n +1 ( λ ) = λu n ( λ ) (iii) the assumption Hyp3 in theorem 3.1 can be replaced byThere exist a sequence γ n that:- | λ | ω n γ n +1 γ n + ω n − ω n γ n +1 γ n − < ; λ ∈ C and n large enough (3 . and- ∞ X k =1 γ n < ∞ (3 . Theorem 3.4.
Let F α the lattice Fock-Bargmann space and H p = M p D p +1 ; p ∈ N the linear unbounded densely defined shift operator acting on F α,p ; p = 0 , , ... the space spanned by e α,νn ( z ) = ( 2 νπ ) / e ν z e − π ν ( n + α ) +2 iπ ( n + α ) z ; n = p, p + 1 , .... } with domain D ( H p ) = { φ ∈ F α ; H p φ ∈ F α } ∩ F α,p whose its adjoint is defined by : H ∗ p e α,νn = ω n,p e α,νn +1 (3 . where A. Intissarn chaoticity of the sum of weighted shift with its adjoint 2013 September 16 ω n,p = ( c α ) (2 p +1) e (2 p +1) µn (3 . with µ := 2 πν and c α = e ( µ +4 α ) and { e α,νn ( z ) = ( 2 νπ ) / e ν z e − π ν ( n + α ) +2 iπ ( n + α ) z ; n = p, p + 1 , .... } is an orthonor-mal basis of F α,p Then we have(i) for all p ≥ , H p is chaotic.(ii) for all p ≥ , H p + H ∗ p is chaotic. Proof (i) In [18], we have shown that for all p ≥ , H p is chaotic.(ii) Let ω n,p = ( c α ) (2 p +1) e (2 p +1) µn ,if we choose now γ n = ( c α ) (2 p +1) e (2 p +1) µβ n with β > | λ | ( c α ) (2 p +1) e (2 p +1) µn γ n +1 γ n + ( c α ) (2 p +1) e (2 p +1) µ ( n − ( c α ) (2 p +1) e (2 p +1) µn γ n +1 γ n − = | λ | ( c α ) (2 p +1) e (2 p +1) µn γ n +1 γ n + e − (2 p +1) µ γ n +1 γ n − = | λ | ( c α ) (2 p +1) e (2 p +1) µn e (2 p +1) µβ + e − (2 p +1) µ (1 − β ) Let m β = | λ | ( c α ) (2 p +1) e (2 p +1) µβ and C β = e − (2 p +1) µ (1 − β ) thenFor β > n ≥ p +1) µ Log m β − C β A. Intissarn chaoticity of the sum of weighted shift with its adjoint 2013 September 16 we deduce that m β e − (2 p +1) µn + C β < . ∞ X n =1 | u n ( λ ) | < ∞ (3 . ∞ X n =1 γ n,p < ∞ the verification of the rest of hypothesis of theorem 3.1 for ω n,p are obvious.This shows that H p + H ∗ p is chaotic for p = 0 , , , ...... Remark 3.5. (i) In each of the above cases the expansion coefficients u n ( λ ) satisfy a three-term recurrence relation of the form u n +1 + α n u n + β n u n − = 0 ( ∗ ) We can consider the above equation as a second order linear homogeneousdifference equation.let us assume that lim α n = α and lim β n = β as n → ∞ . Then thePoincar´e analysis holds see [28] or [29] and states that if t l and t are thetwo roots of the quadratic equation t + αt + β = 0 and | t | > | t | then (*)possesses two linearly independent solutionsThis Poincar´e analysis is not applicable to u n +1 ( λ ) − λω n,p u n ( λ ) + ω n − ,p ω n,p u n − ( λ ) = 0 ( ∗∗ ) we have lim λω n,p = α = 0 and lim ω n − ,p ω n,p = β = 1 as n → ∞ .but the two roots t = i and t = − i of the quadratic equation t + αt + β = 0 have same modulus.(ii) In [23] , we study directly the operator H + H ∗ on classic Bargmannspace and we give the application of theorem 3.1 for H p + H ∗ p on generalizedBargmann space with β = 2 . This last application use the following fondamen-tal lemma. A. Intissarn chaoticity of the sum of weighted shift with its adjoint 2013 September 16
Lemma 3.6.
Let m = 0 , m n = Γ( β ( n +1))Γ( nβ ) ; n = 1 , , ... then m n ∼ ( β ) β n β , n → + ∞ (3 . H p on Fock-Bargmannspace associated to Poincar´e disk In this appendix, we show that the operators H p = A ∗ p A p +1 ; p ∈ N with domain D ( H p ) = { φ ∈ FB ν ; H p φ ∈ FB ν } are chaotic, where FB ν is Fock-Bargmannspace associated to Poincar´e disk, A = ddz and A ∗ = z ddz + 2 νz its adjoint. Theorem 4.1.
Let FB ν = O ( D ) ∩ L ν ( D , dµ ν ( z )); ν > H p = A ∗ p A p +1 ; p ∈ N be the linear unbounded densely defined shift operatoracting on the FB ν,p the space spanned by orthonormal basis P n ( z ) = r (2 ν ) n n ! z n ; n = p, p + 1 , .... } (4 . where (2 ν ) n = Γ(2 ν + n )Γ(2 ν ) is the Pochhammer symbol.with domain D ( H p ) = { φ ∈ FB ν ; H p φ ∈ FB ν } ∩ FB ν,p whose its adjoint is defined by : H ∗ p P n = ω n,p P n +1 (4 . where ω n,p = p ( n + 1)(2 ν + n ) p Y j =1 [( n + 1)(2 ν + n )] ; n ≥ p ≥ . Then we haveFor all p ≥ , H p is chaotic. Proof A. Intissarn chaoticity of the sum of weighted shift with its adjoint 2013 September 16
To use the theorem of B ` es and al, we begin by observing that for φ ( z ) = ∞ X k = p a k P k ( z )such that ∞ X k = p | a k | < ∞ we have the obvious properties(i) H mp φ ( z ) = ∞ X k = p [ m + k − Y j = p ω j,p ] a k + m P k ( z ) of domain D ( H mp ) = { φ = ∞ X k = p a k P k ; ∞ X k = p | a k | < ∞ and ∞ X k = p [ m + k − Y j = p ω j,p ] | a k + m | < + ∞} witch is dense in FB ν,p ∀ m ∈ N (ii) H mp is closed ∀ m ∈ N and H mp P k ( z ) = 0 ∀ m > k ≥ p ≥ ω n,p → + ∞ then the spectrum of H p is the all complex plane.In fact, let φ λ = ∞ X k = p a k P k ( z ) with a k = k − Y j = p λω j,p i.e φ λ = + ∞ X k = p [ k − Y j = p λω j,p ] P k ( z )then as a p = 0 we deduce that H p φ λ = λφ λ , ∀ λ ∈ C . (4 . ∞ X k = p [ k Y j = p λω j,p ] < + ∞ then φ λ ∈ D ( H p )Now, take Y the linear subspace generated by finite combinations of basis { P k } ∞ k = p , this subspace Y is dense in FB ν,p and we define on it the operator S acting on φ = N X k = p a k P k as following S φ = N +1 X k = p a k − ω k − ,p P k (4 . S n P k = 1 Q n + kj = k ω j,p P k + n as n Y j = p ω j,p → + ∞ as n → + ∞ we get S n P k → FB ν,p as n → + ∞ (4 . A. Intissarn chaoticity of the sum of weighted shift with its adjoint 2013 September 16
By noting that H np P k = 0 for n > k and any element of Y can be anni-hilated by a finite power of H p and H p S p = I | Y then the hyperciclycity of H p follows from the theorem of B ` es and al. recalled above.We shall now show that H p has a dense set of periodic points. To see this,it suffices to show that for every element φ in the dense subspace Y there is aperiodic point ψ arbitrarily close to it.For s ≥ p and N ≥ s we put ϕ s,N ( z ) = P s ( z ) + ∞ X k = s +1 [ kN + s − Y j = s ω j,p ] P kN + s ( z ) (4 . Lemma 4.2. (i) H Np kN − Y j =0 ω j,p P kN = ( k − N − Y j =0 ω j,p P ( k − N ∀ k ≥ p (ii) H Np kN − s Y j = s ω j,p P kN + s = ( k − N − Y j = s ω j,p P ( k − N + s for s ≥ p , N ≥ s and k ≥ p (iii) ϕ s,N is N -periodic point of H p .(iv) ϕ s,N ∈ D ( H Np ) . Now, Let φ ( z ) = M X s = p a s P s ( z ) (4 . | a s s − Y j = p ω j,p | < s = p, p + 1 , ........, M (4 . H p as ψ ( z )25 A. Intissarn chaoticity of the sum of weighted shift with its adjoint 2013 September 16 ψ ( z ) = M X s = p a s ϕ s,N ( z ) (4 . N ≥ M such that || φ − ψ ||≤ ǫ ∀ ǫ >
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