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

1.   Introduction

As usual, let ${\mathbb{D}}$ be the unit disk in the complex plane ${\mathbb{C}}$, ${\partial {\mathbb{D}}}$ be the boundary of ${\mathbb{D}}$, $H({\mathbb{D}})$ be the class of functions analytic in ${\mathbb{D}}$ and $H^\infty$ be the set of bounded analytic functions in ${\mathbb{D}}$. Let $0 < p < \infty$. The Hardy space $H^p$ (see ^[5]) is the sets of $f\in H({\mathbb{D}})$ with

Suppose that $K:[0, \infty)\rightarrow[0, \infty)$ is a right-continuous and nondecreasing function with $K(0) = 0$. The Dirichlet Type spaces ${\mathcal{D}_{K}}$, consists of those functions $f\in H({\mathbb{D}})$, such that

The space ${\mathcal{D}_{K}}$ has been extensively studied. Note that $K(t) = t$, it is Hardy spaces $H^2$. When $K(t) = t^{\alpha}$, $0\leq\alpha < 1$, it give the classical weighted Dirichlet spaces $\mathcal{D}_{\alpha}$. For more information on ${\mathcal{D}_{K}}$, we refer to ^{[3,7,8,9,10,14,15,16,19,23]}.

Let $\phi$ be a holomorphic self-map of ${\mathbb{D}}$. The composition operator $C_{\phi}$ on $H({\mathbb{D}})$ 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 $\mathcal{D}_{\alpha}$. El-Fallah, Kellay, Shabankhah and Youssfi ^[7] studied composition operator acting on Dirichlet type spaces $D_{\alpha}^p$ by level set and capacity. For general weighted function $\omega$, Kellay and Lefèvre ^[9] using Nevanlinna type counting functions studied the boundedness and compactness of composition spaces on weighted Hilbert spaces ${\mathcal H}_{\omega}$. After Kellay and Lefèvre's work, Pau and Pérez investigate more properties of composition operators on weighted Dirichlet spaces $\mathcal{D}_{\alpha}$ in ^[14]. For more information on composition operator, we refer to ^[4,18].

We assume that ${\mathcal H}$ is a separable Hilbert space of analytic functions in the unit disc. Composition operator $C_{{\phi}}$ is called power bounded on ${\mathcal H}$ if $C_{{\phi}^n}$ is bounded on ${\mathcal H}$ for all $n\in\mathbb{N}$.

Since power bounded composition operators is closely related to mean ergodic and some special properties (such as: stable orbits) of ${\phi}$, it has attracted the attention of many scholars. Wolf ^[20,21] studied power bounded composition operators acting on weighted type spaces $H_{\upsilon}^{\infty}$. Bonet and Domański ^[1,2] proved that $C_{\phi}$ is power bounded if and only if $C_{\phi}$ is (uniformly) mean ergodic in real analytic manifold (or a connected domain of holomorphy in $\mathcal{C}^d$). Keshavarzi and Khani-Robati ^[11] studied power bounded of composition operator acting on weighted Dirichlet spaces $\mathcal{D}_{\alpha}$. Keshavarzi ^[12] investigated the power bounded below of composition operator acting on weighted Dirichlet spaces $\mathcal{D}_{\alpha}$ 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, ${\mathcal{D}_{K}}$ is the Dirichlet space $\mathcal{D}$. 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 $K(t) = t^p$, $0 < p < 1$, $K(t) = \log\frac{e}{t}$ 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 ${\mathcal{D}_{K}}$. In this paper, the symbol $a\approx b$ means that $a\lesssim b\lesssim a$. We say that $a\lesssim b$ if there exists a constant $C$ such that $a\leq Cb$, where $a, b > 0$.

2.   Power bounded of $C_{{\phi}}$

We assume that ${\mathcal H}$ is a separable Hilbert space of analytic functions in the unit disc. Let $R\in H({\mathbb{D}})$ and $\{R_\zeta: \zeta\in {\mathbb{D}}\}$ be an independent collection of reproducing kernels for ${\mathcal H}$. Here $R_\zeta(z) = R(\bar{\zeta}z)$. The reproducing kernels mean that $f(\zeta) = \langle f, R_\zeta\rangle$ for any $f\in {\mathcal H}$. Let $R_{K, z}$ be the reproducing kernels for ${\mathcal{D}_{K}}$. By ^[3], we see that if $K$ satisfy (1.1) and (1.2), we have $\|R_{K, z}\|_{{\mathcal{D}_{K}}}\approx\frac{1}{\sqrt{K(1-|z|^2)}}$. Before we go into further, we need the following lemma.

Lemma 1. Let $K$ satisfies (1.1) and (1.2). Then

for all $0\leq t < 1$.

Proof. Without loss of generality, we can assume $4/5 < t < 1$. Since $K$ is nondecreasing, we have

Conversely, make change of variables $y = \frac{1}{x}$, an easy computation gives

Let $y = \frac{\gamma}{- \ln t}$. We can deduce that

By ^[6], under conditions (1.1) and (1.2), there exists an enough small $c > 0$ only depending on $K$ such that

and

Therefore,

where $\Gamma(.)$ is the Gamma function. It follows that

The proof is completed.

Theorem 1. Let $K$ satisfy (1.1) and (1.2). Suppose that ${\phi}$ is an analytic selt-map of unit disk which is not the identity or an elliptic automorphism. Then $C_{{\phi}}$ is power bounded on ${\mathcal{D}_{K}}$ if and only if ${\phi}$ has its Denjoy-Wolff point in ${\mathbb{D}}$ and for every $0 < r < 1$, we have

where

and

Proof. Suppose that $w\in{\mathbb{D}}$ is the Denjoy-Wolff point of ${\phi}$ and (A) holds. Then $\lim_{n\rightarrow\infty}{\phi}_n(0) = w$. Hence, there is some $0 < r < 1$ such that $\{{\phi}_n(0)\}_{n\in\mathbb{N}}\subseteq r{\mathbb{D}}$. 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 $C_{{\phi}}$ is power bounded on ${\mathcal{D}_{K}}$. Hence, for any $f\in{\mathcal{D}_{K}}$ and any $n\in\mathbb{N}$, we have $|f({\phi}_n(0))|\lesssim 1.$ Hence, by ^[3], it is easily to see that $\|R_{K, {\phi}_{n}(0)}\|_{{\mathcal{D}_{K}}}\approx\frac{1}{\sqrt{K(1-|{\phi}_n(0)|^2)}}\lesssim 1$. Note that

Therefore, we deduce that ${\phi}_{n}(0)\in r{\mathbb{D}}$, where $0 < r < 1$ and $n\in\mathbb{N}$. Also note that if $w\in\overline{{\mathbb{D}}}$ is the Denjoy-Wolff point of ${\phi}$, we have $\lim_{n\rightarrow\infty}{\phi}_n(0) = w$. Thus, $w\in{\mathbb{D}}$. Let

It is easily to verify that $f_a\in{\mathcal{D}_{K}}$ and $f_a'(z) = \frac{1-|a|^2}{\sqrt{K(1-|a|^2)}(1-\overline{a}z)^2}$. 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 ${\phi}$ 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 $C_{{\phi}}$ is power bounded on ${\mathcal{D}_{K}}$ if and only if

(1). $w\in{\mathbb{D}}.$

(2). $\{{\phi}_n\}$ is a bounded sequence in ${\mathcal{D}_{K}}$.

(3). If $n\in\mathbb{N}$ and $|a|\geq\frac{1+|{\phi}_n(0)|}{2}$, then $\frac{N_{{\phi}_n, K}(a)}{K(1-|a|^2)}\lesssim 1.$

Proof. Suppose that $C_{{\phi}}$ is power bounded on ${\mathcal{D}_{K}}$. By Theorem 1, we see that $w\in{\mathbb{D}}$. Note that $z\in{\mathcal{D}_{K}}$ and ${\phi}_n = C_{{\phi}_n}z$, we have $(2)$ hold. Now, we are going to show (3) hold. Let $|a|\geq\frac{1+|{\phi}_n(0)|}{2}$ and $\Delta(a) = \{z:|z-a| < \frac{1}{2}(1-|a|)\}$. Thus,

If $K$ satisfy (1.1) and (1.2). By ^[9], $N_{\phi_n, K}$ has sub-mean properties. Thus,

Conversely. Suppose that (1)–(3) holds. Let $f\in{\mathcal{D}_{K}}$. Note that $z\in{\mathcal{D}_{K}}$, $z' = 1$ and $\frac{1+|{\phi}_n(0)|}{2} < 1$. By Lemma 1, we see that

Thus,

The proof is completed.

Theorem 3. Let $K$ satisfy (1.1) and (1.2). Suppose that ${\phi}$ is an analytic selt-map of ${\mathbb{D}}$ with Denjoy-Wolff point $w$ and $C_{{\phi}}$ is power bounded on ${\mathcal{D}_{K}}$. Then $f\in{\Gamma_{c, K}({\phi})}$ if and only if for any $\epsilon > 0$,

where ${\Gamma_{c, K}({\phi})} = \{f\in{\mathcal{D}_{K}}: \ C_{{\phi}_n}f \ is\ convergent\}$ and ${\Omega_{\epsilon}(f)} = \{z:(1-|z|^2)^2K(1-|z|^2)|f'(z)|^2\geq\epsilon\}$.

Proof. Let $f\in{\mathcal{D}_{K}}$ and (C) hold. For any $\delta > 0$, we choose $0 < \epsilon < \delta$ and $\epsilon$ is small enough such that

By our assumption, we also know that for this $\epsilon$, there is some $N\in\mathbb{N}$ such that for each $n\geq N$, we have

Since

We obtain

and

Thus,

Conversely. Suppose that $f\in{\mathcal{D}_{K}}$ and $w$ is the Denjoy-Wolff point of ${\phi}$. Thus, $f\circ{\phi}_n\rightarrow f(w)$ uniform convergent and $f\in{\Gamma_{c, K}({\phi})}$ if and only if

Suppose there exist $\epsilon > 0$ such that (C) dose not hold. There is a sequence $\{n_k\}\subseteq\mathbb{N}$ and some $\eta > 0$ such that for any $k\in\mathbb{N}$, we have

Hence,

That is a contradiction. The proof is completed.

3.   Power bounded below of $C_{{\phi}}$

The composition operator $C_{{\phi}}$ is called power bounded below if there exists some $C > 0$ such that $\|C_{{\phi}_n}f\|_{{\mathcal H}}\geq C\|f\|_{{\mathcal H}}$, for all $f\in{\mathcal H}$ and $n\in\mathbb{N}$.

In this section, we are going to show the equivalent characterizations of composition operator $C_{{\phi}}$ power bounded below on ${\mathcal{D}_{K}}$. Before we get into prove, let us recall some notions.

(1) We say that $\{G_n\}$, a sequence of Borel subsets of ${\mathbb{D}}$ satisfies the reverse Carleson condition on ${\mathcal{D}_{K}}$ if there exists some positive constant $\delta$ such that for each $f\in{\mathcal{D}_{K}}$,

(2) We say that $\{\mu_n\}$, a sequence of Carleson measure on ${\mathbb{D}}$ satisfies the reverse Carleson condition, if there exists some positive constant $\delta$ and $0 < r < 1$ such that

for each $a\in{\mathbb{D}}$ and $n\in\mathbb{N}$.

Theorem 4. Let $K$ satisfy (1.1) and (1.2). Suppose that ${\phi}$ is an analytic selt-map of ${\mathbb{D}}$ and $C_{{\phi}}$ is power bounded on ${\mathcal{D}_{K}}$. Then the following are equivalent.

(1). $C_{{\phi}}$ is power bounded below.

(2). There exists some $\delta > 0$ such that $\|C_{{\phi}_n}f_a\|\geq\delta$ for all $a\in{\mathbb{D}}$ and $n\in\mathbb{N}$.

(3). There exists some $\delta > 0$ and $\epsilon > 0$ such that for all $a\in{\mathbb{D}}$ and $n\in\mathbb{N}$,

where $G_{\epsilon}(n) = \{z\in{\mathbb{D}}:\frac{N_{{\phi}_n, K}(z)}{K(1-|z|^2)}\geq\epsilon\}$.

(4). There is some $\epsilon > 0$ such that the sequence of measures $\{\chi_{G_{\epsilon}(n)}dA\}$ satisfies the reverse Carleson condition.

(5). The sequence of measures $\{\frac{N_{{\phi}_n, K}(z)}{K(1-|z|^2)}dA\}$ satisfies the reverse Carleson condition.

(6). There is some $\epsilon > 0$ such that the sequence of Borel sets $\{{G_{\epsilon}(n)}\}$ satisfies the reverse Carleson condition.

Proof. Suppose that $w$ is the Denjoy-Wolff point of ${\phi}$. By Theorem 2, $w\in{\mathbb{D}}$. Without loss of generality, we use $\varphi_w\circ{\phi}\circ\varphi_w$ instead of ${\phi}$.

$(1)\Rightarrow(2)$. It is obvious.

$(3)\Rightarrow(4)$. By ^[6], there exist a small $c > 0$ such that $\frac{K(t)}{t^c}$ 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 $z = \varphi_a(w) = \frac{a-z}{1-\overline{a}z}$, we obtain

Thus,

$(2)\Rightarrow(3)$. Let $r = \sup_{n\in\mathbb{N}}\frac{1+|{\phi}_n(0)|}{2}$. We claim that: there exists some $\epsilon > 0$ and some $\delta > 0$ such that for all $a\in{\mathbb{D}}$ and $n\in\mathbb{N}$,

or

Suppose that there are no $\epsilon, \delta > 0$ such that the above inequalities hold. Thus, there exists sequences $\{a_k\}\subseteq{\mathbb{D}}$ and $\{n_k\}\subseteq\mathbb{N}$ such that

or

Hence,

as $k\rightarrow\infty$. Where

This contradict (2), so our claim hold. Let $\epsilon, \delta > 0$ be as in above. Since $f'_a\rightarrow 0$, uniformly on the compact subsets of ${\mathbb{D}}$, as $|a|\rightarrow 1$, 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 $(3)\Rightarrow(4)$, there must be $\alpha, \beta > 0$ such that

Therefore,

Now if $\{a_k\}$ is a $\alpha$-lattice for ${\mathbb{D}}$, we have

Therefore,

For any $|a|\leq s$, we obtain $|f'_a(z)|\gtrsim(1-s^2)^2K^2(1-s^2)$. Hence,

Therefore, (3) hold.

$(5)\Rightarrow(2)$. Let $a\in{\mathbb{D}}$. Then

$(4)\Rightarrow(6)$. Note that Luecking using a long proof to show that $G$ satisfies the reverse Carleson condition if and only if the measure $\chi_GdA(z)$ is a reverse Carleson measure. Simlar to the proof of ^[13], we omited here.

$(6)\Rightarrow(1)$. Let $f\in{\mathcal{D}_{K}}$. Then

Thus, it is easily to get our result. The proof is completed.

4.   Conclusions

In this paper, we give some equivalent characterizations of power bounded and power bounded below composition operator $C_{{\phi}}$ on Dirichlet Type spaces, which generalize the main results in ^[11,12].

Acknowledgments

The authors thank the referee for useful remarks and comments that led to the improvement of this paper. This work was supported by NNSF of China (No. 11801250, No.11871257), Overseas Scholarship Program for Elite Young and Middle-aged Teachers of Lingnan Normal University, Yanling Youqing Program of Lingnan Normal University, the Key Program of Lingnan Normal University (No. LZ1905), The Innovation and developing School Project of Guangdong Province (No. 2019KZDXM032) and Education Department of Shaanxi Provincial Government (No. 19JK0213).

Conflict of interest

We declare that we have no conflict of interest.