Abstract
In this paper, at first we study boundedness and compactness criterions for generalized composition operator from the Lipschitz space into the Zygmund space. Then we estimate the essential norm of this operator.
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1 INTRODUCTION
Let \(X\) and \(Y\) be Banach spaces. The essential norm of a bounded linear operator \(T:X\to Y\) is its distance to the set of compact operators \(K\) mapping \(X\) into \(Y\), that is,
Let \(\mathbb{D}\) be the open unit disc in the complex plane \(\mathbb{C}\), \(H(\mathbb{D})\) the space of analytic functions on \(\mathbb{D}\) and \(H^{\infty}\) be the space of bounded analytic functions on \(\mathbb{D}\) with norm \(||f||_{\infty}=\sup_{z\in\mathbb{D}}|f(z)|\).
Let \(u\in H(\mathbb{D})\) and \(\varphi\in S(\mathbb{D})\), the set of self-maps of \(\mathbb{D}\). The weighted composition operator with symbols \(u\) and \(\varphi\), denoted by \(uC_{\varphi}\), is defined as follows
where \(M_{u}\) is the multiplication operator with symbol \(u\) and \(C_{\varphi}\) is the composition operator. We refer the interested reader to [4] and [12] for the theory of the composition operators and to [2, 3, 6, 10, 14, 16, 20, 21, 22] for (weighted) composition on various spaces of analytic functions.
Let \(\mathcal{Z}\) denote the set of all functions \(f\in H(\mathbb{D})\cap C(\mathbb{\overline{D}})\) such that
where the supremum is taken over all \(\theta\in\mathbb{R}\) and \(h>0\). By Theorem 5.3 of [12] and the Closed Graph Theorem, we see that an analytic function \(f\) on \(\mathbb{D}\) belongs to \(\mathcal{Z}\) if and only if \(\sup_{z\in\mathbb{D}}(1-|z|^{2})|f^{\prime\prime}(z)|<\infty\). Furthermore,
The preceding quantity is seminorm for the space \(\mathcal{Z}\). The norm defined by
yields a Banach space structure on \(\mathcal{Z}\), which is called the Zygmund space.
Let \(0<\alpha<\infty\). A function \(f\in H(\mathbb{D})\) is said to belong to the Bloch type space \(\mathcal{B}^{\alpha}\) if
Under the seminorm \(f\to\beta_{f}\) , \(\mathcal{B}^{\alpha}\) is conformally invariant, and the norm defined by \(||f||_{\mathcal{B}^{\alpha}}=|f(0)|+\beta_{f}\) yields a Banach space structure on \(\mathcal{B}^{\alpha}\). It is well known that for \(0<\alpha<1\) \(\mathcal{B}^{\alpha}\) is a subspace of \(H^{\infty}\). When \(\alpha=1\), we get the classical Bloch space \(\mathcal{B}\).
The Lipschitz space \(\textrm{Lip}_{\alpha}\) \((\text{with}\quad 0<\alpha<1)\) is the space of functions \(f\in H(\mathbb{D})\) satisfying the Lipschitz condition of order \(\alpha\), i.e, there exists a constant \(C>0\) such that
Such functions \(f\) extend continuously to the closure of the disc. The quantity
defines a norm on \(\textrm{Lip}_{\alpha}\). Let \(f\in\textrm{Lip}_{\alpha}\) and set
Then, for \(z\in\mathbb{D}\), we have \(|f(z)|\leq|f(0)|+C|z|^{\alpha}\leq C|z-w|^{\alpha}\leq||f||_{\textrm{Lip}_{\alpha}}\).
Thus, taking the supremum over \(\mathbb{D}\), we obtain \(||f||_{\infty}\leq||f||_{\textrm{Lip}_{\alpha}}\). By a theorem of Hardy and Littlewood [5], the elements of \(\textrm{Lip}_{\alpha}\) are characterized by the following Bloch-type condition: A function \(f\in H(\mathbb{D})\) belongs to \(\textrm{Lip}_{\alpha}\) if and only if
Moreover,
Composition operators, weighted composition operators, and related operators between the Zygmund space and some various spaces of analytic functions have been studied in [7, 8, 9, 13, 19]. In [8], Li and Stević defined the generalization composition operator \(C_{\varphi}^{g}\) as follows
Li and Stević studied the boundedness and compactness of the generalized composition operator on the Zygmund space and the Bloch type space and the little Bloch type space in [8]. In this paper, we study boundedness and compactness of the generalization composition operator \(C_{\varphi}^{g}\) from \(\textrm{Lip}_{\alpha}\) to \(\mathcal{Z}\). Also we give some estimates for the essential norm of this operator. Weighted composition operators \(uC_{\varphi}\) between \(\textrm{Lip}_{\alpha}\) and \(\mathcal{Z}\) spaces were studied by Colonna and Li in [2]. Some characterizations of the boundedness and compactness of the composition operator, as well as Volterra type operator, on the Bloch type space and the Zygmund space can be found in [1, 5, 17].
The notation \(a\preceq b\) means that there is a positive constant \(C\) such that \(a\leq Cb\). We say that \(a\approx b\) if both \(a\preceq b\) and \(b\preceq a\) hold.
2 BOUNDEDNESS OF THE OPERATOR \(C_{\varphi}^{g}:\textrm{Lip}_{\alpha}\to\mathcal{Z}\)
In this section, we give necessary and sufficient conditions for the boundedness of the operator \(C_{\varphi}^{g}:\textrm{Lip}_{\alpha}\to\mathcal{Z}\).
Theorem 2.1.Let \(0<\alpha<1\), \(g\in H(\mathbb{D})\)and \(\varphi\in S(\mathbb{D})\).Then the operator \(C_{\varphi}^{g}:\textrm{Lip}_{\alpha}\to\mathcal{Z}\)is bounded if only if the following quantities are finite:
and
Proof. For any \(f\in\textrm{Lip}_{\alpha}\),
where in the last inequality we have used (1.1) and the following well known characterization of Bloch type functions (see [18]):
Conversely, assume that \(C_{\varphi}^{g}:\textrm{Lip}_{\alpha}\to\mathcal{Z}\) is bounded. For a fixed \(a\in\mathbb{D}\) and for \(z\in\mathbb{D}\) and \(1\leq j\leq 3\), set
A direct calculation shows that
Then, for \(w\in\mathbb{D}\), we get,
and
Subtracting (2.1) from (2.2), we get
On the other hand, subtracting (2.1) from (2.3), we obtain
Subtracting (2.3) from (2.4), we get
which implies that
Fix \(r\in(0,1)\). If \(|\varphi(w)|>r\), then by (2.5), we have
Taking the functions \(f(z)=z\) and \(f(z)=z^{2}\) respectively, we obtain
and \(N_{2}=\sup_{z\in\mathbb{D}}(1-|z|^{2})|g^{\prime}(z)\varphi(z)+g(z)\varphi^{\prime}(z)|<\infty.\)
If \(|\varphi(w)|<r\), then by (2.6) we get
which, combined with (2.7), implies that \(M_{1}<\infty\). Arguing similarly, we get
and
This means \(M_{2}<\infty\). \(\Box\)
3 COMPACTNESS OF THE OPERATOR \(C_{\varphi}^{g}:\textrm{Lip}_{\alpha}\to\mathcal{Z}\)
In this section, we study the compactness of the operator \(C_{\varphi}^{g}:\textrm{Lip}_{\alpha}\rightarrow\mathcal{Z}\). We begin with the following lemma.
Lemma 3.1 ([11], Lemma 3.7). Let \(0<\alpha<1\)and \(T\)be a bounded linear operator from \(\textrm{Lip}_{\alpha}\)into a normed linear space \(Y\).Then \(T\)is compact if and only if \(||Tf_{n}||_{Y}\rightarrow 0\)whenever \(\{f_{n}\}\)is a norm-bounded sequence in \(\textrm{Lip}_{\alpha}\)that converges to 0 uniformly on \(\bar{\mathbb{D}}\).
Theorem 3.2.Let \(0<\alpha<1\), \(\varphi\in S(\mathbb{D})\)and \(g\in H(\mathbb{D})\).Then \(C_{\varphi}^{g}:\textrm{Lip}_{\alpha}\to\mathcal{Z}\)is compact if and only if bounded,
and
Proof. Let \((z_{k})_{k\in\mathbb{N}}\) be a sequence in \(\mathbb{D}\) such that \(|\varphi(z_{k})|\to 1\) as \(k\to\infty\). Let \(f_{k,j}=\frac{(1-|\varphi(z_{k})|^{2})^{j}}{(1-\overline{\varphi(z_{k}})z)^{j-\alpha}},k\in\mathbb{N}\). Then \(f_{k,j}\in\textrm{Lip}_{\alpha}\), \(\sup_{k\in\mathbb{N}}||f_{k,j}||_{\textrm{Lip}_{\alpha}}<\infty\) and \(f_{k,j}\to 0\) uniformly on \(\mathbb{\overline{D}}\) as \(k\to\infty\). Let \(C_{\varphi}^{g}:\textrm{Lip}_{\alpha}\to\mathcal{Z}\) be compact. By Lemma 3.1 it gives \(\lim_{k\to\infty}||C_{\varphi}^{g}f_{k,j}||_{\mathcal{Z}}=0\). Note that
We have
Consequently,
if one of these two limits exists. Next, set
Then \(h_{k,j}^{\prime}(\varphi(z_{k}))=0\), \(\sup_{k\in\mathbb{N}}||h_{k,j}||_{\mathcal{Z}}<\infty\) and \(h_{k,j}\) converges to \(0\) uniformly on \(\mathbb{\overline{D}}\) as \(k\to\infty\). Since \(C_{\varphi}^{g}:\textrm{Lip}_{\alpha}\to\mathcal{Z}\) is compact, we have \(\lim_{k\to\infty}||C_{\varphi}^{g}h_{k,j}||_{\mathcal{Z}}=0\). On the other hand,
Hence,
Therefore,
This together with (3.2) imply that
Conversely, assume that \(C_{\varphi}^{g}:\textrm{Lip}_{\alpha}\to\mathcal{Z}\) is bounded and (3.1) holds. Since \(C_{\varphi}^{g}:\textrm{Lip}_{\alpha}\to\mathcal{Z}\) is bounded, we have \(||C_{\varphi}^{g}f||_{\mathcal{Z}}\leq C||f||_{\textrm{Lip}_{\alpha}}\) for all \(f\in\textrm{Lip}_{\alpha}\). Taking the functions \(f(z)=z\) and \(f(z)=z^{2}\) respectively, we obtain
and
Using these facts and the boundedness of the function \(\varphi(z)\), we get
Then,
and
On the other hand, from (3.1), for every \(\epsilon>0\), there is a \(\delta\in(0,1)\) such that
whenever \(\delta<|\varphi(z)|<1\). Assume that \((f_{k})_{k\in\mathbb{N}}\) is a sequence in \(\textrm{Lip}_{\alpha}\) such that \(\sup_{k\in\mathbb{N}}||f_{k}||_{\textrm{Lip}_{\alpha}}<\infty\) and \((f_{k})\) converges to \(0\) uniformly on \(\mathbb{\overline{D}}\) as \(k\to\infty\). Let \(U=\{z\in\mathbb{D}:|\varphi(z)|\leq\delta\}\). Then by (3.5) and (3.6), it follows that
So,
The proof is complete. \(\Box\)
4 ESSENTIAL NORM OF \(C_{\varphi}^{g}f:\textrm{Lip}_{\alpha}\to\mathcal{Z}\)
In this section, we give some estimates for the essential norm of operator \(C_{\varphi}^{g}f:\textrm{Lip}_{\alpha}\to\mathcal{Z}\).
Theorem 4.1. Let \(\varphi\in S(\mathbb{D})\) and \(g\in H(\mathbb{D})\) such that \(C_{\varphi}^{g}:\textrm{Lip}_{\alpha}\to\mathcal{Z}\) is bounded. Then
where
Proof. First we prove that \(\max\{{A_{1},A_{2}}\}\leq||C_{\varphi}^{g}||_{e,\textrm{Lip}_{\alpha}\to\mathcal{Z}}\). Let \(a\in\mathbb{D}\). Define
It is easy to check that \(f_{a,j}\in\textrm{Lip}_{\alpha}\) for all \(a\in\mathbb{D}\) and \(f_{a,j}\) converges uniformly to \(0\) on compact subset of \(\textrm{Lip}_{\alpha}\) as \(|a|\to 1\) Thus, for any compact operator \(T:\textrm{Lip}_{\alpha}\to\mathcal{Z}\), we have \(\lim_{|a|\to 1}||Tf_{a,j}||_{\mathcal{Z}}=0,\quad j=1,2\). Hence,
Therefore, based on the definition of the essential norm, we obtain
Now, we prove that \(||C_{\varphi}^{g}f||_{e,\textrm{Lip}_{\alpha}\to\mathcal{Z}}\lesssim\max\{{A_{1},A_{2}}\}.\) For \(r\in[0,1)\), set \(K_{r}:H(\mathbb{D})\to H(\mathbb{D})\) by \((K_{r}f)(z)=f_{r}(z)=f(rz)\). It is obvious that \(f_{r}-f\to 0\) uniformly on compact subsets of \(\mathbb{D}\) as \(r\to 1\). Moreover, the operator \(K_{r}\) is compact on \(\mathcal{B}\) and \(||K_{r}||_{\mathcal{B}\to\mathcal{B}}\leq 1\)(see [10]). By a similar argument it can be proved that the operator \(K_{r}\) is compact on \(\textrm{Lip}_{\alpha}\) and \(||K_{r}||_{\textrm{Lip}_{\alpha}\to\textrm{Lip}_{\alpha}}\leq 1\). Let \(\{r_{j}\}\subset(0,1)\) be a sequence such that \(r_{j}\to 1\) as \(j\to\infty\). Then for all positive integer \(j\), the operator \(C_{\varphi}^{g}K_{r_{j}}:\textrm{Lip}_{\alpha}\to\mathcal{Z}\) is compact. By the definition of the essential norm, we get
For any \(f\in\textrm{Lip}_{\alpha}\) such that \(||f||_{\textrm{Lip}_{\alpha}}\leq 1\),
where \(N\in\mathbb{N}\) is large enough such that \(r_{j}\geq\frac{1}{2}\) for all \(j\in\mathbb{N}\). Since \(C_{\varphi}^{g}:\textrm{Lip}_{\alpha}\to\mathcal{Z}\) is bounded, by (3.3) and (3.4), we have
Since \(r_{j}f_{r_{j}}\to f^{\prime}\) uniformly on compact subsets of \(\mathbb{D}\) as \(j\to\infty\), so
Next we consider \(M_{2}\). We have \(M_{2}\leq\limsup_{j\to\infty}(Q_{1}+Q_{2})\), where
Using the fact that \(||f||_{\textrm{Lip}\alpha}\leq 1\) and (1.1), we obtain
Taking the limit as \(N\to\infty\), we obtain
Similarly,
Hence, we get \(M_{2}\preceq\max\{A_{1},A_{2}\}\). Similarly, it can be shown that \(M_{4}\preceq\max\{A_{1},A_{2}\}\). This completes the proof of the theorem. \(\Box\)
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ACKNOWLEDGMENTS
The authors would like to express their sincere gratitude to the referee for a very careful reading of the paper and for all the valuable suggestions, which led to improvement in this paper.
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Mahmoudfakheh, S., Vaezi, H. Generalized Composition Operators from the Lipschitz Space into the Zygmund Space. J. Contemp. Mathemat. Anal. 55, 283–290 (2020). https://doi.org/10.3103/S1068362320050027
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DOI: https://doi.org/10.3103/S1068362320050027