Strict singularity of weighted composition operators on derivative Hardy spaces

TL;DR

This paper investigates the properties of weighted composition operators on derivative Hardy spaces, establishing a link between strict singularity and compactness, and characterizing operators that fix an isomorphic copy of  for p.

Contribution

It proves that strict singularity coincides with compactness for these operators and characterizes when they fix an isomorphic copy of  on derivative Hardy spaces.

Findings

Strict singularity of the operator is equivalent to its compactness on S^p.

Operators fixing an isomorphic copy of  are characterized for p.

Conditions for weighted composition operators to fix  are provided when p.

Abstract

We prove that the weighted composition operator $W_{\phi,\varphi}$ fixes an isomorphic copy of $\ell^p$ if the operator $W_{\phi,\varphi}$ is not compact on the derivative Hardy space $S^p$. In particular, this implies that the strict singularity of the operator $W_{\phi,\varphi}$ coincides with the compactness of it on $S^p$. Moreover, when $p\neq2$, we characterize the conditions for those weighted composition operators $W_{\phi,\varphi}$ on $S^p$ which fix an isomorphic copy of $\ell^2$ .

Equations

$$\Vert f{\Vert }_{H^p}:={\left(\underset{0\le r<1}{\sup }{\int }_{\partial \mathbb{D}}|f(r\xi ){|}^{p}dm(\xi )\right)}^{1/p}<\infty ,$$

$$\Vert f{\Vert }_{H^p}={\left({\int }_{\partial \mathbb{D}}|f(\xi ){|}^{p}dm(\xi )\right)}^{1/p},$$

$$H^{\infty }=\{f\in H(\mathbb{D}):\Vert f{\Vert }_{\infty }:=\underset{z\in \mathbb{D}}{\sup }\{|f(z)|\}<\infty \}.$$

$$W_{\varphi ,\phi }(f)(z)=\varphi (z)f\circ \phi (z),z\in \mathbb{D},$$

$$S^p=\{f\in H(\mathbb{D}):\Vert f{\Vert }_{S^p}:=|f(0)|+\Vert f^{\prime }{\Vert }_{H^p}<\infty \}.$$

$$\underset{|a|\to 1^{-}}{\lim }{\int }_{\partial \mathbb{D}}\frac{1-|a{|}^2}{|1-\bar{a}\phi (\omega ){|}^2}|\psi (\omega ){\phi }^{\prime }(\omega ){|}^{p}dm(\omega )=0.$$

$$f_a(z)={\int }_0^z\frac{(1-|a{|}^2{)}^{1/p}}{(1-\bar{a}\omega {)}^{2/p}}d\omega ,z\in \mathbb{D},$$

$$E_{\epsilon }:=\{\xi \in L:|1-\phi (\xi )|<\epsilon \}$$

$$\underset{\epsilon \to 0}{\lim }{\int }_{E_{\epsilon }}|(W_{\varphi ,\phi }{f}_{a_n}{)}^{\prime }{|}^{p}dm=0$$

$$\underset{n\to \infty }{\lim }{\int }_{\partial \mathbb{D}\setminus E_{\epsilon }}|(W_{\varphi ,\phi }{f}_{a_n}{)}^{\prime }{|}^{p}dm=0$$

$$\begin{array}{rl}|1-{\bar{a}}_n\phi (\xi )|& =|1-\phi (\xi )+\phi (\xi )-{\bar{a}}_n\phi (\xi )|\\ & \ge |1-\phi (\xi )|-|\phi (\xi )-{\bar{a}}_n\phi (\xi )|\\ & \ge |1-\phi (\xi )|-|1-{\bar{a}}_n|>\frac{\epsilon }{2}.\end{array}$$

$${\int }_{\partial \mathbb{D}\setminus E_{\epsilon }}|(W_{\varphi ,\phi }{f}_{a_n}{)}^{\prime }{|}^{p}dm\le C\left({\int }_{\partial \mathbb{D}\setminus E_{\epsilon }}|{\varphi }^{\prime }{f}_{a_n}(\phi ){|}^{p}dm+{\int }_{\partial \mathbb{D}\setminus E_{\epsilon }}|\varphi {\phi }^{\prime }{f}_{a_n}^{\prime }(\phi ){|}^{p}dm\right).$$

$$\underset{n\to \infty }{\lim }{\int }_{\partial \mathbb{D}\setminus E_{\epsilon }}|{\varphi }^{\prime }{f}_{a_n}(\phi ){|}^{p}dm\le \underset{n\to \infty }{\lim }{\int }_{\partial \mathbb{D}}|W_{{\varphi }^{\prime },\phi }{f}_{a_n}{|}^{p}dm=0.$$

$$\begin{array}{rl}{\int }_{\partial \mathbb{D}\setminus E_{\epsilon }}|\varphi {\phi }^{\prime }{f}_{a_n}^{\prime }(\phi ){|}^{p}dm& ={\int }_{\partial \mathbb{D}\setminus E_{\epsilon }}|\varphi {\phi }^{\prime }{|}^p\frac{1-|a_n{|}^2}{|1-{\bar{a}}_n\phi {|}^2}dm\\ & ={\int }_{\partial \mathbb{D}\setminus E_{\epsilon }}|\varphi {\phi }^{\prime }{|}^p\frac{1-|a_n{|}^2}{|1-{\bar{a}}_n\phi {|}^2}dm\\ & \le \frac{4(1-|a_n{|}^2)}{{\epsilon }^2}{\int }_{\partial \mathbb{D}}|\varphi {\phi }^{\prime }{|}^{p}dm,\end{array}$$

$$\underset{n\to \infty }{\lim }{\int }_{\partial \mathbb{D}\setminus E_{\epsilon }}|\varphi {\phi }^{\prime }{f}_{a_n}^{\prime }(\phi ){|}^{p}dm=0,$$

$$\Vert W_{\varphi ,\phi }(f_{a_n}){\Vert }_{H^p}\ge h>0$$

$$\Vert W_{\varphi ,\phi }(f_{a_n}){\Vert }_{H^p}\ge \Vert \varphi {\phi }^{\prime }{f}_{a_n}^{\prime }(\phi ){\Vert }_{H^p}-\Vert {\varphi }^{\prime }{f}_{a_n}(\phi ){\Vert }_{H^p}.$$

$$\underset{n\to \infty }{\lim }\Vert {\varphi }^{\prime }{f}_{a_n}(\phi ){\Vert }_{H^p}=0.$$

$${\left({\int }_{E_{{\epsilon }_n}}|(W_{\varphi ,\phi }{f}_{b_k}{)}^{\prime }{|}^{p}dm\right)}^{1/p}<4^{-n}\delta h,k=1,\dots ,n-1;$$

$$\begin{split}\|\sum_{j=1}^{\infty}c_{j}W_{\phi,\varphi}(f_{b_{j}})\|^{p}_{S^{p}}&\geq\|\sum_{j=1}^{\infty}\left(c_{j}W_{\phi,\varphi}(f_{b_{j}})\right)^{\prime}\|^{p}_{H^{p}}\\ &=\sum_{n=1}^{\infty}\int_{E_{\varepsilon_{n}}\setminus E_{\varepsilon_{n+1}}}\left|\sum_{j=1}^{\infty}\left(c_{j}W_{\phi,\varphi}(f_{b_{j}})\right)^{\prime}\right|^{p}dm\\ &\geq\sum_{n=1}^{\infty}\Bigg{(}|c_{n}|\left(\int_{E_{\varepsilon_{n}}\setminus E_{\varepsilon_{n+1}}}|\left(W_{\phi,\varphi}(f_{b_{n}})\right)^{\prime}|^{p}dm\right)^{\frac{1}{p}}\\ &\phantom{=}-\sum_{j\neq n}|c_{j}|\left(\int_{E_{\varepsilon_{n}}\setminus E_{\varepsilon_{n+1}}}|\left(W_{\phi,\varphi}(f_{b_{j}})\right)^{\prime}|^{p}dm\right)^{\frac{1}{p}}\Bigg{)}^{p}\,.\end{split}$$

$$\begin{array}{rl}& {\left({\int }_{E_{{\epsilon }_n}\setminus E_{{\epsilon }_{n+1}}}|{\left(W_{\varphi ,\phi }(f_{b_n})\right)}^{\prime }{|}^{p}dm\right)}^{\frac{1}{p}}\\ & ={\left({\int }_{E_{{\epsilon }_n}}|{\left(W_{\varphi ,\phi }(f_{b_n})\right)}^{\prime }{|}^{p}dm-{\int }_{E_{{\epsilon }_{n+1}}}|{\left(W_{\varphi ,\phi }(f_{b_n})\right)}^{\prime }{|}^{p}dm\right)}^{1/p}\\ & \ge {\left({\left(\frac{h}{2}\right)}^p-{\left(4^{-n-1}\delta h\right)}^p\right)}^{1/p}\\ & \ge \frac{h}{2}-4^{-n-1}\delta h\end{array}$$

$${\left({\int }_{E_{{\epsilon }_n}\setminus E_{{\epsilon }_{n+1}}}|{\left(W_{\varphi ,\phi }(f_{b_j})\right)}^{\prime }{|}^{p}dm\right)}^{\frac{1}{p}}<2^{-n-j}\delta h\text{ for }j\ne n.$$

$$\begin{array}{rl}\Vert \sum _{j=1}^{\infty }{c}_j{W}_{\varphi ,\phi }(f_{b_j}){\Vert }_{S^p}& \ge {\left(\sum _{n=1}^{\infty }{\left(|c_n|\left(\frac{h}{2}-4^{-n-1}\delta h\right)-2^{-n}\delta h\Vert (c_j){\Vert }_{{\ell }^p}\right)}^p\right)}^{1/p}\\ & \ge {\left(\sum _{n=1}^{\infty }{\left(|c_n|\left(\frac{h}{2}\right)-2^{-n+1}\delta h\Vert (c_j){\Vert }_{{\ell }^p}\right)}^p\right)}^{1/p}\\ & \ge \frac{h}{2}\Vert (c_j){\Vert }_{{\ell }^p}-\delta h\Vert (c_j){\Vert }_{{\ell }^p}{\left(\sum _{n=1}^{\infty }{2}^{-(n-1)p}\right)}^{1/p}\\ & \ge h\left(\frac{1}{2}-\delta {\left(1-2^{-p}\right)}^{-1/p}\right)\Vert (c_j){\Vert }_{{\ell }^p}\ge C\Vert (c_j){\Vert }_{{\ell }^p},\end{array}$$

$$\Vert \sum _{j=1}^{\infty }{c}_j{W}_{\varphi ,\phi }(f_{b_j}){\Vert }_{S^p}\le C\Vert (c_j){\Vert }_{{\ell }^p}.$$

$$\Vert \sum _{j=1}^{\infty }{c}_j{W}_{\varphi ,\phi }(f_{b_j}){\Vert }_{S^p}=|\sum _{j=1}^{\infty }{c}_j\varphi (0)f_{b_j}(\phi (0))|+\Vert \sum _{j=1}^{\infty }{\left(c_j{W}_{\varphi ,\phi }(f_{b_j})\right)}^{\prime }{\Vert }_{H^p}.$$

$$\Vert \sum _{j=1}^{\infty }{\left(c_j{W}_{\varphi ,\phi }(f_{b_j})\right)}^{\prime }{\Vert }_{H^p}\le C\Vert (c_j){\Vert }_{{\ell }^p}.$$

$$|\sum _{j=1}^{\infty }{c}_j\varphi (0)f_{b_j}(\phi (0))|\le C\Vert (c_j){\Vert }_{{\ell }^1}.$$

$$|\sum _{j=1}^{\infty }{c}_j\varphi (0)f_{b_j}(\phi (0))|\le C\Vert (c_j){\Vert }_{{\ell }^p}.$$

$$C\Vert (c_j){\Vert }_{{\ell }^p}\le \Vert \sum _{j=1}^{\infty }{c}_j{f}_{b_j}{\Vert }_{S^p}\le C\Vert (c_j){\Vert }_{{\ell }^p}.$$

$$\Vert \sum _{j=1}^{\infty }{c}_j{W}_{\varphi ,\phi }(f_{b_j}){\Vert }_{S^p}\ge C\Vert \sum _{j=1}^{\infty }{c}_j{f}_{b_j}{\Vert }_{S^p}$$

$$\Vert W_{\varphi ,\phi }{f}_n{\Vert }_{S^p}>h,$$

Full text

*Corresponding author

This work was supported by NNSF of China (Grant No. 11801094).

1. Introduction

Let $\mathbb{D}$ denote the open unit disk in the complex plane $\mathbb{C}$, and $H(\mathbb{D})$ the space of all analytic functions in $\mathbb{D}$. For $0<p<\infty$, the Hardy space $H^{p}$ is the space of functions $f\in H(\mathbb{D})$ for which

[TABLE]

where $m$ is the normalized Lebesgue measure on $\partial\mathbb{D}$. From [23, Theorem 9.4], this norm is equal to the following norm:

[TABLE]

where for any $\xi\in\partial\mathbb{D}$, $f(\xi)$ is the radial limit which exists almost every.

For $p=\infty$, the space $H^{\infty}$ is defined by

[TABLE]

We define the weighted composition operator $W_{\phi,\varphi}$ for $f\in H(\mathbb{D})$ by

[TABLE]

where $\phi\in H(\mathbb{D})$ and $\varphi$ is an analytic self-map of $\mathbb{D}$. If $\phi(z)\equiv 1$, $W_{\phi,\varphi}$ becomes the composition operator $C_{\varphi}$ while if $\varphi(z)\equiv z$, $W_{\phi,\varphi}$ becomes the multiplication operator $M_{\phi}$. For weighted composition operators $W_{\phi,\varphi}$ on Hardy spaces $H^{p}$, we refer the readers to the literatures [3, 5, 8, 9].

We define the derivative Hardy space $S^{p}$ by

[TABLE]

For $1\leq p\leq\infty$, $S^{p}$ is a Banach algebra and there is an inclusion relation: $S^{p}\subset H^{\infty}$ (for the detail structures of $S^{p}$ spaces, see [6, 7, 12, 15, 16] for references).

In paper [22], Roan started the investigation of composition operators on the space $S^{p}$. After his work, MacCluer [18] gave the characterizations of the boundedness and the compactness of the composition operators on the space $S^{p}$ in terms of Carleson measures. A remarkable result on the boundedness and the compactness of the weighted composition operators on $S^{p}$ was obtained in [2], in which they are both characterized through the corresponding weighted composition operators on $H^{p}$. Furthermore, the isometry between $S^{p}$ was obtained by Novinger and Oberlin in [21], in which they showed that the isometries were closely related to the weighted composition operator.

A bounded operator $T\colon X\to Y$ between Banach spaces is strictly singular if its restriction to any infinite-dimensional closed subspace is not an isomorphism onto its image. This notion was introduced by Kato [13].

A bounded operator $T\colon X\to Y$ between Banach spaces is said to fix a copy of the given Banach space $E$ if there is a closed subspace $M\subset X$, linearly isomorphic to $E$, such that the restriction $T_{|M}$ defines an isomorphism from $M$ onto $T(M)$. The bounded operator $T\colon X\to Y$ is called $\ell^{p}$-singular if it does not fix any copy of $\ell^{p}$ .

Laitila, et al [14] recently investigated the strict singularity for the composition operators on $H^{p}$ spaces. Following their ideas, Miihkinen [19] studied the strict singularity of $T_{g}$ on Hardy space $H^{p}$ and showed that the strict singularity of $T_{g}$ coincides with its compactness on $H^{p},\,1\leq p<\infty.$ Miihkinen [19] also post an open question which was resolved in [20] by utilizing the generalized Volterra operators. It should be noticed that Hernández, et al [11] investigated the interpolation and extrapolation of strictly singular operators between $L_{p}$ spaces.

In this paper, we prove that the weighted composition operator $W_{\phi,\varphi}$ fixes an isomorphic copy of $\ell^{p}$ if the operator $W_{\phi,\varphi}$ is not compact on the derivative Hardy space $S^{p}$. In particular, this implies that the strict singularity of the operator $W_{\phi,\varphi}$ coincides with the compactness of it on $S^{p}$. Moreover, when $p\neq 2$, we characterize the conditions for those weighted composition operators $W_{\phi,\varphi}$ on $S^{p}$ which fix an isomorphic copy of $\ell^{2}$ .

Our main results read as follows:

Theorem 1**.**

Let $1\leq p<\infty$, $\phi\in H(\mathbb{D})$ and $\varphi$ is an analytic self-map of $\mathbb{D}$. If the weighted composition operator $W_{\phi,\varphi}\colon S^{p}\to S^{p}$ is bounded but not compact, then $W_{\phi,\varphi}$ fixes an isomorphic copy of $\ell^{p}$ in $S^{p}$. In particular, the operator $W_{\phi,\varphi}$ is not strictly singular, that is, strict singularity of bounded operator $W_{\phi,\varphi}$ coincides with its compactness.

Remark 1**.**

In the final section, we prove that the claims of theorem 1 is still true for the case of $p=\infty$ .

Denote $E_{\varphi}=\{\zeta\in\partial\mathbb{D}:|\varphi(\zeta)|=1\}$, then we have

Theorem 2**.**

Let $1\leq p<\infty$, $\phi\in H(\mathbb{D})$ and $\varphi$ is an analytic self-map of $\mathbb{D}$. Suppose that $W_{\phi,\varphi}\colon S^{p}\to S^{p}$ is bounded and $m(E_{\varphi})=0$. If $W_{\phi,\varphi}$ is bounded below on an infinite-dimensional subspace $M\subset S^{p}$, then the restriction $W_{\phi,\varphi}$ on $M$ fixes an isomorphic copy of $\ell^{p}$ in $M$. In particular, if $p\neq 2$, the operator $W_{\phi,\varphi}$ does not fix any isomorphic copy of $\ell^{2}$ in $S^{p}$.

When $m(E_{\varphi})>0$, it holds that

Theorem 3**.**

Let $1\leq p<\infty$, $\phi\in H(\mathbb{D})$ and $\varphi$ is an analytic self-map of $\mathbb{D}$. Suppose that $W_{\phi,\varphi}\colon S^{p}\to S^{p}$ is bounded. if $m(E_{\varphi})>0$ and $\phi\varphi^{\prime}\neq 0$, then the operator $W_{\phi,\varphi}$ fixes an isomorphic copy of $\ell^{2}$ in $S^{p}$.

Notation: throughout this paper, $C$ will represents a positive constant which may be given different values at different occurrences.

2. Proof of Theorem 1

This section is devoted to the proof of Theorem 1. First, the following Lemma 1 can be deduced from [2, Theorem 2.1] and [9, Theorem 2.2 and Theorem 2.3].

Lemma 1**.**

Let $1\leq p<\infty$, $\phi\in H(\mathbb{D})$ and $\varphi$ is an analytic self-map of $\mathbb{D}$. Then $W_{\phi,\varphi}\colon S^{p}\to S^{p}$ is compact if and only if $\phi\in S^{p}$ and

[TABLE]

The following lemma 2 is proven in [2, Proposition 3.3(ii)].

Lemma 2**.**

Let $1\leq p\leq\infty$, $\phi\in H^{p}$ and $\varphi$ is an analytic self-map of $\mathbb{D}$. Then $W_{\phi,\varphi}\colon S^{p}\to H^{p}$ is compact.

We employ the test functions

[TABLE]

where $a\in\mathbb{D}$. They all satisfy $\|f_{a}\|_{S^{p}}=1$ and $f_{a}$ converges to [math] uniformly on compact subsets of $\mathbb{D}$, as $|a|\to 1^{-}$ .

Let $L=\{\xi\in\partial\mathbb{D}:\ \text{the radial limit }\varphi(\xi)\ \text{exists}\}$ and

[TABLE]

for any given $\varepsilon>0$, then $m(\partial\mathbb{D}\setminus L)=0$. The proof of Theorem 1 relies on the following auxiliary lemma.

Lemma 3**.**

Let $(a_{n})\subset\mathbb{D}$ be a sequence such that $0<|a_{1}|<|a_{2}|<\ldots<1$ and $a_{n}\to 1$. If the bounded operator $W_{\phi,\varphi}\colon S^{p}\to S^{p}$ is not compact, then we have

[TABLE]

Proof.

(1) For each fixed $n$, this follows immediately from the absolute continuity of Lebesgue measure, the boundedness of operator $W_{\phi,\varphi}$ and the fact that $W_{\phi,\varphi}$ is not compact (which implies that $\varphi$ is not identically $1$) .

(2) For any given $\varepsilon>0$, let $\xi\in L\setminus E_{\varepsilon}$. Then there exists an $N>0$ such that whenever $n>N$, it holds that

[TABLE]

Now, by definition, we have

[TABLE]

Since $W_{\phi,\varphi}$ is bounded, it follows that $\phi\in S^{p}$, that is, $\phi^{\prime}\in H^{p}$. By Lemma 2, $W_{\phi^{\prime},\varphi}:\ S^{p}\to H^{p}$ is compact, which implies that

[TABLE]

For the estimate of the second integral, we have

[TABLE]

where $\int_{\partial\mathbb{D}}|\phi\varphi^{\prime}|^{p}dm$ is finite due to the boundedness of $W_{\phi,\varphi}:S^{p}\to S^{p}$ and [2, Theorem 2.1] and [8, Theorem 4].

Therefore,

[TABLE]

The proof is complete. ∎

Now, we are ready to give a proof of Theorem 1.

Proof of Theorem 1.

First, we prove that there exists a sequence $(a_{n})\subset\mathbb{D}$ with $0<|a_{1}|<|a_{2}|<\ldots<1$ and $a_{n}\to\omega\in\partial\mathbb{D},$ such that there is a positive constant $h$ such that

[TABLE]

holds for all $n\in\mathbb{N}$ .

Since $W_{\phi,\varphi}:S^{p}\to S^{p}$ is not compact, Lemma 1, there exists a sequence $(a_{n})\subset\mathbb{D}$ with $0<|a_{1}|<|a_{2}|<\ldots<1$ and $a_{n}\to\omega\in\partial\mathbb{D},$ such that there is a positive constant $h$ such that $\|\phi\varphi^{\prime}f^{\prime}_{a_{n}}(\varphi)\|_{H^{p}}\geq 2h>0$ holds for all $n\in\mathbb{N}$ . Note that

[TABLE]

By Lemma 2, $W_{\phi^{\prime},\varphi}:\ S^{p}\to H^{p}$ is compact, which implies that

[TABLE]

Hence, there exists a subsequence of $(a_{n})$ (denoted still by $(a_{n})$) such that the above claim holds. We assume without loss of generality that $a_{n}\rightarrow 1$ as $n\rightarrow\infty$ by utilizing a suitable rotation.

Then by Lemma 3 and induction method, we are able to choose a decreasing positive sequence $(\varepsilon_{n})$ such that $E_{\varepsilon_{1}}=\partial\mathbb{D}$ and $\lim_{n\rightarrow\infty}\varepsilon_{n}=0$, and a subsequence $(b_{n})\subset(a_{n})$ such that the following three conditions hold:

[TABLE]

for every $n\in\mathbb{N},$ where $\delta>0$ is a small constant whose value will be determined later.

Now we are ready to prove that there is a $C>0$ such that the inequality $\|\sum_{j=1}^{\infty}c_{j}W_{\phi,\varphi}(f_{b_{j}})\|_{S^{p}}\geq C\|(c_{j})\|_{\ell^{p}}$ holds. By the triangle inequality in $L^{p}$, we have

[TABLE]

Observe that for every $n\in\mathbb{N},$ we have

[TABLE]

according to conditions (1) and (3) above, where the last estimate holds for $1\leq p<\infty$.

Moreover, by condition (1) and (2), it holds that

[TABLE]

Consequently, we obtain that

[TABLE]

where the last inequality holds when we choose $\delta$ small enough.

On the other hand, we are to prove the converse inequality:

[TABLE]

By definition,

[TABLE]

First, we note that a straightforward variant of the above procedure also gives

[TABLE]

Next, when $p=1$, since $\lim_{j\rightarrow\infty}f_{b_{j}}(\varphi(0))=0$, it is trivial that

[TABLE]

When $1<p<\infty$, we can choose a subsequence of $(b_{j})$ (still denoted by $(b_{j})$) such that $\{(1-|b_{j}|^{2})^{1/p}\}_{j=1}^{\infty}\in\ell^{q}$, where $1/p+1/q=1$ . Then by Hölder’s inequality,

[TABLE]

Accordingly, the desired inequality follows.

By choosing $\phi=1$ and $\varphi=z$, we obtain that

[TABLE]

Thus, we have

[TABLE]

The proof is complete. ∎

3. Proof of Theorem 2

In this section, we give the proof of Theorem 2.

Proof of Theorem 2.

Since $M$ is the infinite-dimensional subspace of $S^{p}$ and polynomials are dense in $S^{p}$ (see [15]), there exists a sequence $(f_{n})$ of unit vectors in $M$ such that $f_{n}$ converges to [math] uniformly on compact subsets of $\mathbb{D}$. since $W_{\phi,\varphi}$ is bounded below on $M\subset S^{p}$, there exists $h>0$ such that

[TABLE]

for all $n\in\mathbb{N}$ . For $k\geq 1$, denote $E_{k}:=\{\xi\in\partial\mathbb{D}:\ |\varphi(\xi)|\geq 1-1/k\}$ . Since by assumption, $\lim_{k\to\infty}m(E_{k})=m(E_{\varphi})=0$, it holds that

[TABLE]

for every $n\in\mathbb{N}$. Moreover, since $f_{n}$ converges to [math] uniformly on compact subsets of $\mathbb{D}$, it follows that

[TABLE]

for every $k\in\mathbb{N}.$

The remainder of the proof is an argument that goes exactly as the proof of Theorem 1, so we omit it. Thus, the proof is complete. ∎

4. Proof of Theorem 3

In this last section, we give a proof for Theorem 3.

Proof of Theorem 3.

We define the subspace $S^{p}_{0}$ of $S^{p}$ by

[TABLE]

Then the integral operator $T_{z}:\ f\mapsto\int_{0}^{z}f(\zeta)d\zeta$ is an isometric isomorphism from $H^{p}$ onto $S^{p}_{0}$ . Then the weighted composition operator $W_{\phi,\varphi}$ on $S^{p}_{0}$ is unitary similar to the operator

[TABLE]

By Lemma 2, [2, Theorem 2.1] and the expression of the operator $T$, we see that $W_{\phi,\varphi}$ is bounded on $S^{p}$ if and only if $\phi\in S^{p}$ and $W_{\phi,\varphi}$ is bounded on $S^{p}_{0}$ .

Now, for the operator $W_{\phi\varphi^{\prime},\varphi}$ on $H^{p}$, we can deduce from the proof of [17, Theorem 2] that there exists a sequence of integers $(n_{k})$ satisfying $\inf_{k}(n_{k+1}/n_{k})>1$ and a positive constant $C$ such that

[TABLE]

where $e_{n_{k}}:=z^{n_{k}}$ is the unit vector in $H^{p}$. Since the operator $W_{\phi^{\prime},\varphi}\circ T_{z}:H^{p}\to H^{p}$ is compact (it is equivalent to the compactness of $W_{\phi^{\prime},\varphi}:S^{p}\to H^{p}$, which is claimed by Lemma 2), then for any $\varepsilon>0$, there exists a subsequence of $(n_{k})$ (still denoted as $(n_{k})$) such that

[TABLE]

Thus,

[TABLE]

which implies that the weighted composition operator $W_{\phi,\varphi}$ on $S^{p}_{0}$ is bounded below:

[TABLE]

where $g_{n_{k}}:=T_{z}(e_{n_{k}})$ is the unit vector in $S^{p}_{0}$ .

Since Paley’s theorem (see [10]) implies that the closed linear span $M:=\{e_{n_{k}}:k\geq 1\}$ in $H^{p}$ is isomorphic to $\ell^{2}$, which implies that the closed linear span $T_{z}(M)=\{g_{n_{k}}:k\geq 1\}$ in $S^{p}_{0}$ is isomorphic to $\ell^{2}$ . Hence, $W_{\phi,\varphi}$ fixes an isomorphic copy of $\ell^{2}$ in $S^{p}_{0}$ .

Accordingly, it follows that $W_{\phi,\varphi}$ fixes an isomorphic copy of $\ell^{2}$ in $S^{p}$ since $S^{p}_{0}\subset S^{p}$ and $\|W_{\phi,\varphi}\|_{S^{p}}\geq\|W_{\phi,\varphi}\|_{S^{p}_{0}}$, which is the desired result. ∎

5. The strict singularity of $W_{\phi,\varphi}$ on $S^{\infty}$

Here we show that the claims of theorem 1 is still true for the case of $p=\infty$ . We have known that the weighted composition operator $W_{\phi,\varphi}$ on $S^{\infty}_{0}$ is unitary similar to the operator

[TABLE]

It follows from [4] that any weakly compact weighted composition operator on $H^{\infty}$ is compact. Since by Lemma 2 the operator $W_{\phi\varphi^{\prime},\varphi}$ is compact on $H^{\infty}$, it holds that $T$ is weakly compact on $H^{\infty}$ is and only if $T$ is compact on $H^{\infty}$ . Moreover, Bourgain [1] established that a bounded linear operator on $H^{\infty}$ is weakly compact if and only if it does not fix any copy of $\ell^{\infty}$ . Thus, $T$ is compact on $H^{\infty}$ if and only if it does not fix any copy of $\ell^{\infty}$. Therefore, the weighted composition operator $W_{\phi,\varphi}$ on $S^{\infty}_{0}$ is compact if and only if it does not fix any copy of $\ell^{\infty}$.

By Lemma 2, [2, Theorem 2.1] and the expression of the operator $T$, we see that $W_{\phi,\varphi}$ is bounded on $S^{\infty}$ if and only if $\phi\in S^{\infty}$ and $W_{\phi,\varphi}$ is bounded on $S^{\infty}_{0}$ . Moreover, under the assumption for the boundedness of $W_{\phi,\varphi}$ on $S^{\infty}$, $W_{\phi,\varphi}$ is bounded on $S^{\infty}$ if and only if $W_{\phi,\varphi}$ is bounded on $S^{\infty}_{0}$ .

Therefore, the weighted composition operator $W_{\phi,\varphi}$ on $S^{\infty}$ is compact if and only if it does not fix any copy of $\ell^{\infty}$. In particular, the noncompact operator $W_{\phi,\varphi}$ on $S^{\infty}$ is not strictly singular, that is, strict singularity of bounded operator $W_{\phi,\varphi}$ on $S^{\infty}$ coincides with its compactness.