Devaney Chaotic And Distributional Chaotic Properties Of Operator

Definition 1 Let X be a topology vector space , then an operator T : X→X is called hypercyclic if there is a vector x∈X whose orbit under Tis dense in X.definition 2 Let X be a topology vector space , then an operator T : X→X is called (topologically) transitive if for each pair U and V of non-empty open subsets of x there is some m∈N with T^m(U)∩V≠0 .Hypercyclicity is closely related to the concept of transitivity from topology dynamics .Theorem 1 ( Birkhoff Transitivity Theorem ) Let X be a separable F-space. Then an operator T : X→X is hypercyclic if and only if it is transitive .A discrete dynamical system is simply a continuous mapping f : X→X , where X is a complete separable metric space .definition 3 Let < X , f > be a dynamical system. f is Devaney chaotic if(1)f is transitive;(2) the periodic points for f are dense in X; and(3) f has sensitive dependence on initial conditions. (1) +(2) implies (3).We now introduce Devaney chaotic operator .Theorem 2 Let X be an F-sequence space in which , (e_n)_n∈N is an uncondition basis . Let T : X→X be a unilateral weighted backward shift with weight sequence (a_n)_n∈N. Then the following assertions are equivalent: (1) T is Devaney chaotic ;(2) T is hypercyclic and has a non-trivial periodic point;(3) T has a non-trivial periodic point;(4) the series (?) converges in X .Theorem 3 Let X be a bilateral F-sequence space in which , (e_n)_n∈N is an uncondition basis . Let T : X→X be a bilateral weighted backward shift with weight sequence (a_n)_n∈NThen the following assertions are equivalent:(1) T is Devaney chaotic ;(2) T is hypercyclic and has a non-trivial periodic point;(3) T has a non-trivial periodic point;(4) the series (?)Cowen-Douglas operator is an important kind of operators , there is a good condition for it is Devaney chaotic .Definition 4 ForΩa connected open subset of C and n a positive integer, let B_n(Ω) denotes the operators T in L(H) which satisfy:(a)Ω(?)σ(T) = {ω∈C : T -ωnot invertible};(b) ran(T -ω) = H forωinΩ;(c) (?) ker_ω∈Ω(T -ω) =H and(d) dim ker(T -ω) = n forωinΩ..Theorem 4 Let T∈B_n(Ω). IfΩ∩S≠Φ, then T is Devaney chaotic.Distributionally chaotic is anther kind definition of chaos in dynamic system .Distributional chaos is defined in the following way.For any pair {x,y} (?) X and any n∈N, define distributional function F_xyⁿ : R→[0,1]:Furthermore, define Both F_xy and F_xy^* are nondecreasing functions and may be viewed as cumulative probability distributional functions satisfying F_xy(Ï„) = F_xy^*(Ï„) = 0 forÏ„< 0.Definition 5 {x,y} (?) X is said to be a distributionally chaotic pair, ifFurthermore, f is called distributionally chaotic, if there exists an uncountable subset (?) X such that each pair of two distinct points in A is a distributionally chaotic pair.The norm-unimodal operator is a criterion for a bounded linear operator be a distributional chaotic .Theorem 5 (Distributionally Chaotic Criterion) Let X be a Banach space and let T∈L(X). If T is norm-unimodal, then T is distributionally chaotic.Theorem 6 (Weakly Distributionally Chaotic Criterion) Let X be a Banach space and let T∈L(X). If for any sequence of positive numbers C_m increasing to +∞, there exist {x_m}_m=1^∞in X satisfying(1) (?)= 0.(2) There is a sequence of positive integers N_m increasing to +∞, such that (?)1Then T is distributionally chaotic.There are many operators are not chaotic .Proposition 1 Let N be a normal operator on separable complex Hilbert space. Then N is impossible to be Li-Yorke chaotic. Consequently, N is neither Devaney nor normuniodal nor distributionally chaotic .Proposition 2 Let K be a compact operator on complex Hilbert space, then K is neither Devaney nor norm-unimodal nor distributionally chaotic .Let K be a compact operator on complex Hilbert space, (?)λ∈C. ThenλI + K is not norm-unimodal , but can be distributional chaotic .Proposition 3 Let K be a compact operator on complex Hilbert space, A∈C. ThenλI + K is not norm-unimodal.Proposition 4 For any e > 0, there is a small compact operator ||K_ε||< e such that I + K_εis distributionally chaotic. Proposition 5 Given a nest algebra T(N), there exists an distributional operator.Proposition 6 Given a nest N , then there exist an operator T in T(N) such that T is distributionally chaotic.