Power bounded and power bounded below composition operators on Dirichlet Type spaces

: Motivated by [11, 12], under some conditions on weighted function K , we investigated power bounded and power bounded below composition operators on Dirichlet Type spaces D K


Introduction
As usual, let D be the unit disk in the complex plane C, ∂D be the boundary of D, H(D) be the class of functions analytic in D and H ∞ be the set of bounded analytic functions in D. Let 0 < p < ∞.The Hardy space H p (see [5]) is the sets of f ∈ H(D) with Suppose that K : [0, ∞) → [0, ∞) is a right-continuous and nondecreasing function with K(0) = 0.The Dirichlet Type spaces D K , consists of those functions f ∈ H(D), such that The space D K has been extensively studied.Note that K(t) = t, it is Hardy spaces H 2 .When K(t) = t α , 0 ≤ α < 1, it give the classical weighted Dirichlet spaces D α .For more information on D K , we refer to [3, 7-10, 14-16, 19, 23].
Let φ be a holomorphic self-map of D. The composition operator C φ on H(D) is defined by It is an interesting problem to studying the properties related to composition operator acting on analytic function spaces.For example: Shapiro [17] introduced Nevanlinna counting functions studied the compactness of composition operator acting on Hardy spaces.Zorboska [23] studied the boundedness and compactness of composition operator on weighted Dirichlet spaces D α .El-Fallah, Kellay, Shabankhah and Youssfi [7] studied composition operator acting on Dirichlet type spaces D p α by level set and capacity.For general weighted function ω, Kellay and Lefèvre [9] using Nevanlinna type counting functions studied the boundedness and compactness of composition spaces on weighted Hilbert spaces H ω .After Kellay and Lefèvre's work, Pau and Pérez investigate more properties of composition operators on weighted Dirichlet spaces D α in [14].For more information on composition operator, we refer to [4,18].We assume that H is a separable Hilbert space of analytic functions in the unit disc.Composition operator C φ is called power bounded on H if C φ n is bounded on H for all n ∈ N.
Since power bounded composition operators is closely related to mean ergodic and some special properties (such as: stable orbits) of φ, it has attracted the attention of many scholars.Wolf [20,21] studied power bounded composition operators acting on weighted type spaces H ∞ υ .Bonet and Domański [1,2] proved that C φ is power bounded if and only if C φ is (uniformly) mean ergodic in real analytic manifold (or a connected domain of holomorphy in C d ).Keshavarzi and Khani-Robati [11] studied power bounded of composition operator acting on weighted Dirichlet spaces D α .Keshavarzi [12] investigated the power bounded below of composition operator acting on weighted Dirichlet spaces D α later.For more results related to power bounded composition operators acting on other function spaces, we refer to the paper cited and referin [1,2,11,12,20,21].
We always assume that K(0) = 0, otherwise, D K is the Dirichlet space D. The following conditions play a crucial role in the study of weighted function K during the last few years (see [22]): and where Note that the weighted function K satisfies (1.1) and (1.2), it included many special case, such as t and so on.Some special skills are needed in dealing with certain problems.Motivated by [11,12], using several estimates on the weight function K, we studying power bounded composition operators acting on D K .In this paper, the symbol a ≈ b means that a b a.We say that a b if there exists a constant C such that a ≤ Cb, where a, b > 0.

Power bounded of C φ
We assume that H is a separable Hilbert space of analytic functions in the unit disc.Let R ∈ H(D) and {R ζ : ζ ∈ D} be an independent collection of reproducing kernels for H.Here R ζ (z) = R( ζz).The reproducing kernels mean that f (ζ) = f, R ζ for any f ∈ H. Let R K,z be the reproducing kernels for D K .By [3], we see that if K satisfy (1.1) and (1.2), we have . Before we go into further, we need the following lemma.Lemma 1.Let K satisfies (1.1) and (1.2).Then Proof.Without loss of generality, we can assume 4/5 < t < 1.Since K is nondecreasing, we have Conversely, make change of variables y = 1 x , an easy computation gives dy.
Let y = γ − ln t .We can deduce that By [6], under conditions (1.1) and (1.2), there exists an enough small c > 0 only depending on Therefore, where Γ(.) is the Gamma function.It follows that .
The proof is completed.
Theorem 1.Let K satisfy (1.1) and (1.2).Suppose that φ is an analytic selt-map of unit disk which is not the identity or an elliptic automorphism.Then C φ is power bounded on D K if and only if φ has its Denjoy-Wolff point in D and for every 0 < r < 1, we have Proof.Suppose that w ∈ D is the Denjoy-Wolff point of φ and (A) holds.Then lim n→∞ φ n (0) = w.Hence, there is some 0 < r < 1 such that {φ n (0)} n∈N ⊆ rD.Thus, From [24], we see that Let {a i } be a r-lattice.By sub-mean properties of | f |, combine with (B), we deduce Thus, On the other hand.Suppose that C φ is power bounded on D K .Hence, for any f ∈ D K and any n ∈ N, we have | f (φ n (0))| 1.Hence, by [3], it is easily to see that R K,φ n (0 1.

Note that lim
Therefore, we deduce that φ n (0) ∈ rD, where 0 < r < 1 and n ∈ N. Also note that if w ∈ D is the Denjoy-Wolff point of φ, we have lim n→∞ φ n (0) = w.Thus, w ∈ D. Let .
It is easily to verify that f a ∈ D K and f a (z) Thus, combine with (B), we have Thus, (A) hold.The proof is completed.Theorem 2. Let K satisfy (1.1) and (1.2).Suppose that φ is an analytic selt-map of unit disk which is not the identity or an elliptic automorphism with w as its Denjoy-wolff point.Then C φ is power bounded on D K if and only if (1).w ∈ D.
Proof.Suppose that C φ is power bounded on D K .By Theorem 1, we see that w ∈ D. Note that z ∈ D K and φ n = C φ n z, we have ( 2 If K satisfy (1.1) and (1.2).By [9], N φ n ,K has sub-mean properties.Thus, Thus, The proof is completed.
Theorem 3. Let K satisfy (1.1) and (1.2).Suppose that φ is an analytic selt-map of D with Denjoy-Wolff point w and C φ is power bounded on D K .Then f ∈ Γ c,K (φ) if and only if for any > 0, Proof.Let f ∈ D K and (C) hold.For any δ > 0, we choose 0 < < δ and is small enough such that By our assumption, we also know that for this , there is some N ∈ N such that for each n ≥ N, we have We obtain and Thus, Conversely.Suppose that f ∈ D K and w is the Denjoy-Wolff point of φ.Thus, Suppose there exist > 0 such that (C) dose not hold.There is a sequence {n k } ⊆ N and some η > 0 such that for any k ∈ N, we have Hence, That is a contradiction.The proof is completed.

Power bounded below of C φ
The composition operator C φ is called power bounded below if there exists some C > 0 such that In this section, we are going to show the equivalent characterizations of composition operator C φ power bounded below on D K .Before we get into prove, let us recall some notions.
(1) We say that {G n }, a sequence of Borel subsets of D satisfies the reverse Carleson condition on D K if there exists some positive constant δ such that for each (2) We say that {µ n }, a sequence of Carleson measure on D satisfies the reverse Carleson condition, if there exists some positive constant δ and 0 < r < 1 such that µ n (D(a, r)) > δ|D(a, r)| for each a ∈ D and n ∈ N. Theorem 4. Let K satisfy (1.1) and (1.2).Suppose that φ is an analytic selt-map of D and C φ is power bounded on D K .Then the following are equivalent.
(1).C φ is power bounded below.(2).There exists some δ > 0 such that C φ n f a ≥ δ for all a ∈ D and n ∈ N.
Proof.Suppose that w is the Denjoy-Wolff point of φ.By Theorem 2, w ∈ D. Without loss of generality, we use ϕ w • φ • ϕ w instead of φ.
(3) ⇒ (4).By [6], there exist a small c > 0 such that K(t) t c is nondecreasing (0 < t < 1).Thus, the proof is similar to [18, page 5].Let 0 < r < 1 and C > 0 such that Making change of variable z = ϕ a (w) = a−z 1−az , we obtain Thus, ( . We claim that: there exists some > 0 and some δ > 0 such that for all a ∈ D and n ∈ N, Suppose that there are no , δ > 0 such that the above inequalities hold.Thus, there exists sequences This contradict (2), so our claim hold.Let , δ > 0 be as in above.Since f a → 0, uniformly on the compact subsets of D, as |a| → 1, there exists some 0 < s < 1 such that for all |a| > s, we have That is, for |a| > s, we deduce that Similar to the proof of (3) ⇒ (4), there must be α, β > 0 such that  (4) ⇒ (6).Note that Luecking using a long proof to show that G satisfies the reverse Carleson condition if and only if the measure χ G dA(z) is a reverse Carleson measure.Simlar to the proof of [13], we omited here.
(6) ⇒ (1).Let f ∈ D K .Then Thus, it is easily to get our result.The proof is completed.

Conclusions
In this paper, we give some equivalent characterizations of power bounded and power bounded below composition operator C φ on Dirichlet Type spaces, which generalize the main results in [11,12].