Proof The principal arguments below are modeled after [Reference Shapiro10].

(i) $\Rightarrow $ (ii) For $n\in \mathbb N$ , let

$$\begin{align*}K_{\Theta, n}(\mathbb D) = \{ f\in K_\Theta(\mathbb D): f\ \textrm{has zero of order}\ n\ \textrm{at the origin}\}. \end{align*}$$

Let $P_n: K_{\Theta }(\mathbb D)\to K_{\Theta , n}(\mathbb D)$ denote the orthogonal projector. We claim that

(3.2) $$ \begin{align} \|C_\varphi P_n\|_{K_{\Theta}(\mathbb D) \to H^2(B_d)}\to 0 \end{align} $$

as $n\to \infty $ . Then $C_\varphi $ is compact, since it is approximable by the finite-rank operators $C_\varphi (I-P_n)$ .

To verify (3.2), fix $\varepsilon>0$ and $f\in K_{\Theta }(\mathbb D)$ , $\|f\|\le 1$ . Let $g_n = P_n f$ , then $\|g_n\|\le 1$ .

First, let $p\in (0,1)$ be that provided by Theorem 2.1. By (3.1),

(3.3) $$ \begin{align} \int_{\partial B_d}N^p_{\varphi_\zeta}(w) \frac{(1-|\Theta(w)|)^p}{(1-|w|)^p}\, d\sigma_d(\zeta)\to 0 \quad \textrm{as}\ |w|\to 1-. \end{align} $$

For $r\in (0,1)$ , put

$$\begin{align*}\mathfrak{N}_{p, r} = \sup_{r< |w| <1} \int_{\partial B_d} N_{\varphi_\zeta}(w)\frac{(1-|\Theta(w)|)^p}{1-|w|}\, d\sigma_d(\zeta). \end{align*}$$

By Littlewood’s inequality [Reference Littlewood8],

$$\begin{align*}N_{\varphi_\zeta}(w) \le \log\left|\frac{1-\overline{w}\varphi(0)}{\varphi(0) -w}\right|,\quad w\in\mathbb D\setminus \{\varphi(0)\}. \end{align*}$$

Thus, for $\frac {1+|\varphi (0)|}{2} < |w| <1$ , we obtain $N_{\varphi _\zeta }(w) \le C (1-|w|)$ , hence,

$$\begin{align*}N^{1-p}_{\varphi_\zeta}(w) \le C (1-|w|)^{1-p}, \quad \frac{1+|\varphi(0)|}{2} < |w| <1. \end{align*}$$

Therefore,

(3.4) $$ \begin{align} \mathfrak{N}_{p, r}\to 0 \quad \textrm{as}\ \ r\to 1- \end{align} $$

by (3.3). Now, using (3.4) and applying (2.2) to $g_n \in {K_{\Theta }}$ , $\|g_n\|\le 1$ , choose R so close to $1$ that

(3.5) $$ \begin{align} \int_{\mathbb D \setminus R\mathbb D} \frac{|g_n^\prime(w)|^2}{(1-|\Theta(w)|)^p} \mathfrak{N}_{p, R} (1-|w|)\, dA(w) < C \|g_n\| \mathfrak{N}_{p, R} < \varepsilon \end{align} $$

for all $n\in \mathbb N$ .

Second,

$$\begin{align*}\max_{|w|<R}|g^\prime_n(w)|\to 0 \end{align*}$$

as $n\to \infty $ . Thus,

(3.6) $$ \begin{align} \int_{R\mathbb D} {|g_n^\prime(w)|^2}\int_{\partial B_d} N_{\varphi_\zeta}(w)\, d\sigma_d(\zeta)\, dA(w) < \varepsilon \end{align} $$

for all sufficiently large n.

By (2.4),

$$ \begin{align*} \frac12 \|C_\varphi g_n\|_{H^2(B_d)}^2 & = \int_{\mathbb D} |g_n^\prime(w)|^2 \int_{\partial B_d} N_{\varphi_\zeta}(w)\, d\sigma_d(\zeta)\, dA(w) + |g_n(\varphi(0))|^2\\ & =\int_{\mathbb D\setminus R\mathbb D} + \int_{R\mathbb D} +\ |g_n(\varphi(0))|^2. \end{align*} $$

Observe that $|g_n(\varphi (0))|^2 \to 0$ as $n\to \infty $ . Hence, combining (3.5) and (3.6), we conclude that

$$\begin{align*}\|C_\varphi g_n\|_{H^2(B_d)}\to 0, \end{align*}$$

as required.

(ii) $\Rightarrow $ (i) Assume that (i) does not hold. Then, there exists a sequence $\{w_n\}_{n=1}^\infty \subset \mathbb D$ such that $|w_n|\to 1$ and

(3.7) $$ \begin{align} \int_{\partial B_d}N_{\varphi_\zeta}(w_n) \frac{1-|\Theta(w_n)|}{1-|w_n|}\, d\sigma_d(\zeta)\ge c> 0. \end{align} $$

Sequentially applying (2.4), Lemma 2.6(ii), and Corollary 2.5, we obtain

(3.8) $$ \begin{align} \begin{aligned} \|C_\varphi k_{w_n}(z)\|^2/\|k_{w_n}\|^2 &\ge \frac{1}{\|k_{w_n}\|^2} \int_{\mathbb D} |k^\prime_{w_n}(z)|^2 \left(\int_{\partial B_d} N_{\varphi_\zeta}(z)\, d\sigma_d(\zeta) \right)\, dA(z)\\ &\ge \int_{D(w_n, \varepsilon)} \frac{C}{(1- |w_n|^2)^{3}} \left(\int_{\partial B_d} N_{\varphi_\zeta}(z)\, d\sigma_d(\zeta) \right)\, dA(z)\\ &\ge \frac{C_\varepsilon}{1- |w_n|^2} \int_{\partial B_d} N_{\varphi_\zeta}(w_n)\,d\sigma_d(\zeta). \end{aligned} \end{align} $$

Now, recall that $C_\varphi $ is a compact operator by (ii), thus, $\|C_\varphi k_{w_n}(z)\|/\|k_{w_n}\| \to 0$ by Lemma 2.6(i). Therefore, (3.8) contradicts (3.7). The proof of the theorem is finished.

Proof As indicated in [Reference Lyubarskii and Malinnikova9, Section 4], the results of Baranov [Reference Baranov2] on compact Carleson embeddings of the model spaces $K^p_\Theta $ guarantee that for a one-component inner function $\Theta $ , the compactness of the composition operator $C_\varphi : K^p_\Theta (\mathbb D)\to H^p(\mathbb D)$ does not depend on $p\in (1, \infty )$ . Finally, for $p=2$ , it suffices to apply Theorem 1 from [Reference Lyubarskii and Malinnikova9] or Theorem 3.1.

Proof By Theorem 3.1, it suffices to show that the compactness of $C_\varphi : K^p_\Theta (\mathbb D)\to H^p(B_d)$ does not depend on $p\in (1, \infty )$ . To prove this property, assume that $p_0\in (1, \infty )$ and $C_\varphi : K^{p_0}_\Theta (\mathbb D)\to H^{p_0}(B_d)$ is a compact operator. We are planning to prove that $C_\varphi : K^{p}_\Theta (\mathbb D)\to H^{p}(B_d)$ is also compact for all $p \in (1, \infty ) \setminus \{p_0\}$ . In fact, for definiteness and without loss of generality, we may assume that $p> p_0$ .

Fix p and $p_1$ such that $p_1> p > p_0$ . Define $\theta \in (0,1)$ by the following identity:

(4.1) $$ \begin{align} \frac{1}{p} = \frac{1-\theta}{p_0} + \frac{\theta}{p_1}. \end{align} $$

On the one hand, (4.1) guarantees that $(K_\Theta ^{p_0}, K_\Theta ^{p_1})_{\theta , p} = K_\Theta ^{p}$ (see, e.g., [Reference Kislyakov and Zlotnikov7] for more general results even in the bidisk $\mathbb D^2$ ). On the other hand, it is known that $(H^{p_0}(B_d), H^{p_1}(B_d))_{\theta , p} = H^{p}(B_d)$ , since $p_1> p_0 > 1$ . Indeed, application of the classical Riesz projection reduces interpolation between $H^{p_0}(B_d)$ and $H^{p_1}(B_d)$ to that between $L^{p_0}$ and $L^{p_1}$ .

Now, observe that $C_\varphi : K^{p_1}_\Theta (\mathbb D)\to H^{p_1}(B_d)$ is a bounded operator. Indeed, as indicated in the introduction, $C_\varphi : H^q(\mathbb D) \to H^q(B_d)$ is bounded for all $0<q<\infty $ . Thus, applying Theorem 4.2 for $q=p$ and the pairs $(K_\Theta ^{p_0}, K_\Theta ^{p_1})$ and $(H^{p_0}(B_d), H^{p_1}(B_d))$ , we conclude that $C_\varphi : K^{p}_\Theta (\mathbb D)\to H^{p}(B_d)$ is compact, as required.