Chaotic Properties Of Hilbert Space Operators

A discrete dynamical system is simply a continuous mapping f: X→X where X is a complete separable metric space.In 1975, Li and Yorke [1] observed complicated dynamical behavior for the class of interval maps with period 3. This phenomena is currently well known under the name of Li-Yorke chaos. Therefrom, several kinds of chaos were well studied.Definition 1 {x, y} (?) X is said to be a Li-Yorke chaotic pair, ifFurthermore, f is called Li-Yorke chaotic, if there exists an uncountable subsetΓ(?) X such that each pair of two distinct points inΓis a Li-Yorke chaotic pair.From Schweizer and Smftal's paper [2], 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, defineBoth 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 2 {x, y} (?) X is said to be a distributionally chaotic pair, if Furthermore, 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. Moreover, (?) is called a distributionallyε-scrarnbled set.Distributional chaos always implies Li-Yorke chaos, as it requires more complicated statistical dependence between orbits than the existence of points which are proximal but not asymptotic. The converse implication is not true in general. However in practice, even in the simple case of Li-Yorke chaos, it might be quite difficult to prove chaotic behavior from the very definition. Such attempts have been made in the context of linear operators (see [3,4]). Further results of [3] were extended in [7] to distributional chaos for the annihilation operator of a quantum harmonic oscillator. More about distributional chaos, one can see [5, 6, 8, 9,10].From Rolewicz's article [18], hypercyclicity is widely studied. In fact, it coincides a dynamical property "transitivity". Now there has been got so many improvements at this aspect (Grosse-Erdmann's and Shapiro's articles [16, 17] are good surveys.). Specially, distributionalchaos for shift operators were discussed by F. Martinez-Gim(?)nez, et.al. in [11]. In a recent article [13], Hou Bingzhe introduces a new dynamical property for linear operators called norm-ummodality which implies distributional chaos and studies chaotic properties for Cowen-Douglas operators. Later, Hou Bingzhe, Tian Geng and Shi Luoyi obtain some properties of norm-unimodel operators and consider distributionally chaotic property of normaloperators, subnormal operators and compact perturbation ofλI (see [14]).First, we give some results of [14] as follows.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 not distributionally chaotic.Corollary 1 Let T be a subnormal operator on separable complex Hilbert space. Then T is impossible to be Li-Yorke chaotic. Consequently, T is not distributionally chaotic.Proposition 2 Let if be a compact operator on separable complex Hilbert space, then K is impossible to be Li-Yorke chaotic. Consequently, K is not distributional chaotic. Similarly one can gainλI + K (|λ|≠1) is impossible to be Li-Yorke chaotic. Consequently,it's not distributional chaotic. But when |λ| = 1, there is not a general conclusion. Hou Bingzhe obtain,Proposition 3 For anyε> 0, there is a small compact operator K_ε, such that ||K_ε|| <εand I + K_εis distributionally chaotic.Next, we consider the interior point of DC(H) and LY(H).Theorem DC(H)⁰ = LY(H)⁰ = {T∈B(H), (?)λ∈(?)D s.t. ind(λ-T)> 0}.Parallelly one can consider the closure of DC(H) and LY(H).