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A matrix acting between Fock spaces
Journal of Inequalities and Applications volume 2024, Article number: 8 (2024)
Abstract
If \(\mathcal{H}_{\nu}=(\nu _{n,k})_{n,k\geq 0}\) is the matrix with entries \(\nu _{n,k}=\int _{[0,\infty )}\frac{ t^{n+k}}{n!}\,d\nu (t)\), where ν is a nonnegative Borel measure on the interval \([0,\infty )\), the matrix \(\mathcal{H}_{\nu}\) acts on the space of all entire functions \(f(z) =\sum_{n=0}^{\infty} a_{n} z^{n}\) and induces formally the operator in the following way:
In this paper, for \(0< p\leq \infty \), we classify for which measures the operator \(\mathcal{H}_{\nu}(f)\) is well defined on \(F^{p}\) and also gets an integral representation, and among them we characterize those for which \(\mathcal{H}_{\nu}\) is a bounded (resp., compact) operator between \(F^{p} \) and \(F^{\infty }\).
1 Introduction
Throughout this paper, we write \(A \lesssim B\) for nonnegative quantities A and B if there exists a constant C (independent of A and B) such that \(A \leq CB\). The symbol \(A \simeq B\) means that both \(A \lesssim B \) and \(B \lesssim A\). C denotes a finite constant that may change value from one occurrence to the next.
Let \(\mathbb{C}\) be the complex plane and \(H(\mathbb{C})\) be the class of all entire functions. If \(0< p <\infty \), then the Fock space \(F^{p} \) is the set of all \(f\in H(\mathbb{C})\) such that
where \(z = x+iy\) and \(dA(z) = dx\,dy\) is the Lebesgue area measure on \(\mathbb{C}\). Set
In particular, \(F^{2} \) is a reproducing kernel Hilbert space. The function \(K_{z}(w)=e^{ z \overline{w}}\) is the reproducing kernel for \(F^{2} \) and
is the normalized kernel. Moreover, each \(k_{a}\) is a unit vector in \(F^{p}\), where \(0< p\leq \infty \). Let \(f^{\infty }\) denote the space of entire functions such that
Interested readers can refer to [18] for the theory of Fock spaces.
Let \(0< p, q<\infty \), and μ be a nonnegative Borel measure on \(\mathbb{C}\). Recall that μ is a \((p, q)\)-Fock Carleson measure if the identity operator i is bounded from \(F^{p}\) to \(L^{q}(e^{-\frac{q}{2}|\cdot |^{2}}\,d\mu )\), i.e., there exists some constant C such that, for all \(f \in F^{p}\),
When \(p=q\), μ is exactly the Fock Carleson measure for \(F^{p} \) (see [11, 18]). Also, μ is called a vanishing \((p, q)\)-Fock Carleson measure if
whenever \(\{f_{j} \}\) is a bounded sequence in \(F^{p}\) that converges to 0 uniformly on compact subsets of \(\mathbb{C} \) as \(j \rightarrow \infty \).
The matrix \(\mathcal{H}_{\nu}= (\nu _{n,k})_{n,k\geq 0}\) induces formally an operator (which will also be denoted by \(\mathcal{H}_{\nu}\)) on \(H(\mathbb{C})\) in the following sense. For any \(f(z) =\sum_{n=0}^{\infty} a_{n} z^{n}\in H(\mathbb{C})\), by multiplication of the matrix with the sequence of Taylor coefficients of the function, we can define
If the right-hand side makes sense and defines a function in \(H(\mathbb{C})\), then the matrix \(\mathcal{H}_{\nu}\) induces formally an operator \(\mathcal{H}_{\nu}\) on \(H(\mathbb{C})\).
Let \(\mathbb{D}\) be the open unit disc and \(H(\mathbb{D})\) be all of the analytic functions on \(\mathbb{D}\). If we replace \(H(\mathbb{C})\) by \(H(\mathbb{D})\) in the definition of operators above, there is a rich history of these operators (which will be denoted by \(H_{\nu}\)) on several natural \(L^{p}\) spaces of analytic functions on \(\mathbb{D}\), especially those on the Hardy space and the Bergman space. For example, the Hilbert operator induced by the Hilbert matrix \((\frac{1}{n+k+1})_{n,k\geq 0}\) has been studied in Hardy spaces [5] and Bergman spaces [4]; estimates on the norms have also been obtained. More generally, another approach to the study of \(H_{\nu}\) on spaces of analytic functions is developed. Galanopoulos and Peláez introduced Hankel matrix \(H_{\nu}= (\int _{[0,1)}t^{n+k}\,d\nu (t) )_{n,k\geq 0}\), where ν is a finite nonnegative Borel measure on \([0,1)\), and also investigated the boundedness and compactness of the operator induced by \(H_{\nu}\) on the Hardy space \(H^{1}\) and the Bergman space \(A^{2}\) in [6]. Chatzifountas, Girela, and Peláez [2] later generalized the operator \(H_{\nu}\) on Hardy spaces \(H^{p}\). In [7, 8], Girela and Merchán also studied the operator \(H_{\nu}\) acting on certain möbius invariant spaces on the unit disk such as the Bloch space, BMOA, the analytic Besov spaces, etc. Recently, Ye and Zhou considered a new operator, called derivative-Hilbert operator, induced by some Hankel matrix on analytic function spaces on \(\mathbb{D}\) in [16, 17]. It turns out that the derivative-Hilbert operator and the Hilbert operator \(H_{\nu}\) are closely related. For more results on the operator induced by some Hankel matrix, we refer to [1, 5, 14, 15].
From now on, ν denotes a nonnegative Borel measure on \([0,\infty )\). In the Fock space setting, Zhuo et al. [19] considered a special matrix \(\mathcal{H}_{\nu}= (\nu _{n,k})_{n,k\geq 0}\) with entries
and characterized those nonnegative Borel measures ν such that the operator \(\mathcal{H}_{\nu}\) was well defined on the Fock space \(F^{p} \) (\(0< p<\infty \)). Furthermore, \(\mathcal{H}_{\nu}\) was rewritten as
Under this integral representation of \(\mathcal{H}_{\nu}\), the measures ν for which \(\mathcal{H}_{\nu}\) is a bounded (resp., compact) operator from the Fock space \(F^{p} \) into \(F^{q} \) (\(0< p,q<\infty \)) were characterized. The main purpose of this paper is to extend the operator \(\mathcal{H}_{\nu}\) to be well defined on the Fock space \(F^{\infty}\). With \(0 < p \leq \infty \), we are going to obtain some characterizations on those measures ν such that the operator \(\mathcal{H}_{\nu}\) is bounded or compact from \(F^{p} \) to \(F^{\infty }\) and from \(F^{\infty }\) to \(F^{p}\), respectively.
2 Preliminaries
In this section, we state some lemmas for the proof of our main results. The following lemma can be found in [13].
Lemma 2.1
Suppose \(\alpha > 0\). For every positive integer n,
The following formula is used many times throughout this paper. See [18, Corollary 2.5] for a proof.
Lemma 2.2
Suppose \(\alpha > 0\) and β is real. Then
for all \(w\in \mathbb{C}\).
Lemma 2.3
Let \(f(z)=\sum a_{n} z^{n}\) be an entire function. Then
Proof
By Cauchy’s integral formula and Hölder’s inequality, it is easy to see that
for every \(n \in \mathbb{N}\) and \(r > 0\), and hence \(\|a_{n}z ^{n}\|_{\infty}\lesssim \|f\|_{\infty}\). Note that, by elementary calculations and Stirling’s formula,
Thus, for every n,
□
Let \(D(z, r) \) denote the Euclidean disk centered at z with radius r. The following basic estimate for integral averages of functions in Fock spaces can be found in [18, Lemma 2.32].
Lemma 2.4
For any \(0< p<\infty \) and \(r\in (0,\infty )\), there exists a positive constant \(C =C(p,r)\) such that, for all \(z\in \mathbb{C}\),
for all entire functions f.
We also need the following result. See [18, Corollary 2.8] for a proof.
Lemma 2.5
Let \(0< p\leq \infty \) and \(f\in F^{p} \). Then
for all \(z\in \mathbb{C}\).
Given \(r > 0\), a sequence \(\{a_{k}\} \) in \(\mathbb{C}\) is called an r-lattice if \(\bigcup^{\infty}_{k=1}D(a_{k},r)\) covers \(\mathbb{C}\) and the disks \(\{D(a_{k},r/3)\}^{\infty}_{k=1}\) are pairwise disjoint. For any \(\delta > 0\), there exists a positive integer m (depending only on r and δ) such that every point in \(\mathbb{C}\) belongs to at most m of the sets \(D(a_{k},\delta )\), see [18, page 118]. The following three lemmas characterize the \((p,q)\)-Fock Carleson measure and vanishing \((p,q)\)-Fock Carleson measure for \(0< p,q<\infty \), which can be found in [9]. For a Borel measure μ on \(\mathbb{C}\), define \(\widehat{\mu}_{r}(z)=\frac{\mu (D(z,r))}{\pi r^{2}}\). Let \(t>0\), the t-Berezin transform of μ is defined by
whenever these integrals converge, see [18, page 120].
Lemma 2.6
Let \(0< p\leq q<\infty \), and let \(\mu \geq 0\). Then the following statements are equivalent:
-
(1)
μ is a \((p,q)\)-Fock Carleson measure;
-
(2)
\(\widehat{\mu}_{r}(z)\) is bounded on \(\mathbb{C}\) for some (or any) \(r>0\);
-
(3)
\(\tilde{\mu}_{t}(z)\) is bounded on \(\mathbb{C}\) for some (or any) \(t>0\). Furthermore,
$$ \Vert i \Vert _{ F^{p} \to L^{q} (e^{-\frac{q}{2} \vert \cdot \vert ^{2}}\,d\mu )}\simeq \bigl\Vert \widehat{ \mu}_{r}^{\frac{1}{q}} \bigr\Vert _{L^{\infty}}\simeq \bigl\Vert \tilde{\mu}_{t}^{ \frac{1}{q}} \bigr\Vert _{L^{\infty}}. $$
Lemma 2.7
Let \(0< p\leq q<\infty \), and let \(\mu \geq 0\). Then the following statements are equivalent:
-
(1)
μ is a vanishing \((p,q)\)-Fock Carleson measure;
-
(2)
\(\widehat{\mu}_{r}(z)\to 0\) as \(z \to \infty \) for some (or any) \(r>0\);
-
(3)
\(\tilde{\mu}_{t}(z)\to 0\) as \(z \to \infty \) for some (or any) \(t>0\).
Lemma 2.8
Let \(0< q< p<\infty \) and let \(\mu \geq 0\). Set \(s=\frac{p}{q}\) and \(s'\) to be the conjugate exponent of s. Then the following statements are equivalent:
-
(1)
μ is a \((p,q)\)-Fock Carleson measure;
-
(2)
μ is a vanishing \((p,q)\)-Fock Carleson measure;
-
(3)
\(\widehat{\mu}_{r}(z)\in L^{s'}(dA)\) for some (or any) \(r>0\).
In the light of above three lemmas, the notion of (vanishing) \((p,q)\)-Fock Carleson measures does not depend on the particular value of p, q, but depends only on the ratio \(s=\frac{p}{q}\) in the case \(0 < q < p < \infty \). Let \(\Lambda ^{s}\) be the class of all \((p,q)\)-Fock Carleson measures such that \(s=\frac{p}{q}\) and \(\Lambda ^{s}_{0}\) be the class of all vanishing \((p,q)\)-Fock Carleson measures such that \(s=\frac{p}{q}\). When \(0 < s\leq 1\) (equivalently, \(p \leq q\)), we simply write Λ and \(\Lambda _{0}\) for \(\Lambda ^{s}\) and \(\Lambda ^{s}_{0}\) respectively. That is,
and
Notice that \(\Lambda ^{s}\subset \Lambda \) and \(\Lambda ^{s}_{0}\subset \Lambda _{0}\) for all \(s > 0\).
3 Conditions such that \(\mathcal{H}_{\nu}\) is well defined on \(F^{p}\)
In this section, we first clarify for which measures the operator \(\mathcal{H}_{\nu}\) is well defined on Fock spaces and also gets an integral representation. Throughout the paper, a nonnegative Borel measure μ on \([0,\infty )\) can be seen as a Borel measure on \(\mathbb{C}\) by identifying it with the measure \(\mu ^{\star}\) defined by
for any Borel subset A of \(\mathbb{C}\). In this way a positive Borel measure μ on \([0,\infty )\) is an a \((p,q)\)-Fock Carleson measure if and only if there exists a positive constant C such that
for any \(a\in [0,\infty )\) and any fixed \(0 \leq r<\infty \).
Theorem 3.1
Suppose \(0 < p\leq \infty \) and ν is a nonnegative Borel measure on \([0, \infty )\). If \(e^{\epsilon |\cdot |^{2}}\nu \in \Lambda \) for any fixed \(\epsilon >\frac{1}{2}\), then the power series in (1.1) is a well defined entire function for every \(f\in F^{p} \). Furthermore,
Proof
The case when \(0 < p <\infty \) has been proved in [19]. We now consider the case \(p =\infty \). Suppose that \(e^{\epsilon |\cdot |^{2}}\nu \in \Lambda \) with any fixed \(\epsilon >\frac{1}{2}\), it might as well assume that \(e^{\epsilon |\cdot |^{2}}\nu \) is \((1,1)\)-Fock Carleson measure for any \(0< s<\infty \) by Lemma 2.5. For any \(0 < r < \infty \), fix \(f(z)=\sum_{n=0}^{\infty} a_{n} z^{n} \in F^{\infty} \) and \(|z| \leq r\). We deduce that, by Lemma 2.2,
So the integral in (3.1) uniformly converges for every domain \(\{z:|z| \leq r\}\), the resulting function is analytic in \(\mathbb{C}\) and, for every \(z \in \mathbb{C}\),
By Lemma 2.6, we may assume that \(e^{\epsilon |\cdot |^{2}}\nu \) is \((2,2)\)-Fock Carleson measure without loss of generality. This together with Hölder’s inequality and Lemma 2.1 shows that
Combining this with Lemma 2.3, we conclude that for every n,
Therefore, the series in (1.1) is well defined for all \(z \in \mathbb{C}\), and
By (3.2), we obtain
This proves the desired result. □
Theorem 3.2
Suppose that ν is a nonnegative Borel measure on \([0, \infty ) \) such that
Then the power series in (1.1) is well defined for every \(f\in F^{\infty} \) and (3.1) holds.
Proof
Assume that ν satisfies (3.3) and fix \(f(z)=\sum_{n=0}^{\infty} a_{n} z^{n} \in F^{\infty} \). From the definition of \(\nu _{n,k}\), it is elementary to check that
The same arguments show that
On the other hand, for any \(n \in \mathbb{N}\),
However, by Hölder’s inequality, we have
It follows from Lemma 2.3 and (3.3) that the series \(\sum_{k=0}^{\infty}a_{k}\nu _{n,k}\) is absolutely convergent, and
Furthermore,
for each \(z\in \mathbb{C}\). This shows that the power series in (1.1) represents an entire function and
The proof is completed. □
We also obtain the following necessary condition for the operator \(\mathcal{H}_{\nu}\) to be well defined on Fock spaces.
Theorem 3.3
Let ν be a nonnegative Borel measure on \([0, \infty )\). If the integral
converges absolutely for every \(f\in F^{\infty} \), then \(e^{\frac{1}{2}|\cdot |^{2}}\,d\nu \) is a \((\infty ,1)\)-Fock Carleson measure.
Proof
For every \(f\in F^{\infty} \), the integral \(\int _{[0, \infty )}f(t)e^{ t z}\,d\nu (t)\) converges absolutely for \(z=0\), we have
By the closed graph theorem, the identity mapping is bounded from \(F^{\infty} \) into \(L^{1}(e^{\frac{1}{2}|\cdot |^{2}}\,d\nu )\), which implies the desired estimate. □
Specializing to the space \(f^{\infty}\), we use duality theorem for \(f^{\infty}\) to obtain the following result. Recall that \((f^{\infty})^{*}= F^{1}\) under the pairing
We refer the interested reader to [18, Theorem 2.26 and page 39] for details.
Theorem 3.4
Let dν be a nonnegative Borel measure on \([0,\infty )\). If for any \(f\in f^{\infty}\) the integral
converges absolutely, then
Proof
By assumption, taking \(z = 0\), there is \(C>0\) such that
for all \(r \in (0,\infty )\). More specifically, choosing \(f=1\), we have \(\int _{[0, \infty )}\,d\nu (t)<\infty \), which means dν is a finite Borel measure. On the other hand, using Hölder’s inequality, we obtain that
Since the set of finite linear combinations of kernel functions is dense in \(f^{\infty} \) [18, Lemma 2.11] and all kernel functions belong to \(F^{2}\), the reproducing property and Fubini’s theorem imply that for any \(f\in f^{\infty} \),
where \(g_{r}(z)=\int _{[0, r)}e^{ t z}\,d\nu (t)\). By the duality relation \((f^{\infty})^{*}= F^{1}\) and the uniform boundedness principle, we get \(g_{r}\in F^{1}\) and \(\sup_{r}\|g_{r}\|_{1}< C\). Since \(F^{1}\subset f^{\infty}\), we replace f by \(g_{r}\) in (3.4) and obtain that
However, Lemma 2.5 implies that
Combining this with the previous inequality and letting \(r\to \infty \), we have
This proves the desired result. □
4 The operator \(\mathcal{H}_{\nu}\) acting between Fock spaces
In this section, for \(0 < p\leq \infty \), we are going to characterize those measures ν for which \(\mathcal{H}_{\nu}\) is a bounded (resp., compact) operator from \(F^{p} \) to \(F^{\infty }\) or from \(F^{\infty }\) to \(F^{p}\) for some \(0< p\leq \infty \). Now, we state the main results as follows.
Theorem 4.1
Suppose \(0 < p\leq \infty \). Let ν be a nonnegative Borel measure on \([0, \infty )\) such that \(e^{\epsilon |\cdot |^{2}}\nu \in \Lambda \) with any fixed \(\epsilon >\frac{1}{2}\). Then \(\mathcal{H}_{\nu}\) is bounded from \(F^{p} \) into \(F^{\infty}\) if and only if \(e^{ |\cdot |^{2}}\nu \in \Lambda \).
Proof
By Theorem 3.1, \(\mathcal{H}_{\nu}\) has an integral representation (3.1). Suppose that \(\mathcal{H}_{\nu}\) is a bounded operator from \(F^{p} \) into \(F^{\infty }\). Fixed \(r>0\), Lemmas 2.4 and 2.2 show that there is \(C > 0\) such that
for any \(a\in [0, \infty )\). This proves that \(e^{ |\cdot |^{2}}\nu \in \Lambda \) by Lemma 2.6.
Conversely, suppose \(e^{ |\cdot |^{2}}\nu \in \Lambda \). Given \(f\in F^{p}\), by Lemmas 2.5, 2.2, and 2.6, we have
for any \(z\in \mathbb{C}\). Therefore, \(\mathcal{H}_{\nu}\) is bounded from \(F^{p} \) into \(F^{\infty }\). This completes the proof of the theorem. □
Lemma 4.1
Suppose that \(0 < p,q \leq \infty \) and \(\mathcal{H}_{\nu}\) is bounded from \(F^{p} \) into \(F^{q} \). Then \(\mathcal{H}_{\nu}\) is a compact operator if and only if for any bounded sequence \(\{f_{n}\}\) in \(F^{p} \), which converges uniformly to 0 on every compact subset of \(\mathbb{C}\), we have \(\mathcal{H}_{\nu}(f_{n})\to 0\) as \(n\to 0\) in \(F^{q} \).
The proof of Lemma 4.1 is similar to that of Proposition 3.11 in [3]. We omit the details.
Theorem 4.2
Suppose \(0 < p\leq \infty \). Let ν be a nonnegative Borel measure on \([0, \infty )\) such that \(e^{\epsilon |\cdot |^{2}}\nu \in \Lambda \) with any fixed \(\epsilon >\frac{1}{2}\). Then \(\mathcal{H}_{\nu}\) is a compact operator from \(F^{p} \) into \(F^{\infty }\) if and only if \(e^{ |\cdot |^{2}}\nu \in \Lambda _{0}\).
Proof
Assume that \(\mathcal{H}_{\nu}\) is a compact operator from \(F^{p} \) into \(F^{\infty }\). It is easy to see that the function \(k_{a}(z)\to 0\) uniformly on compact sets as \(a \to \infty \). Using Lemma 4.1, we obtain that \(\{\mathcal{H}_{\nu}(k_{a})\}\) converges to 0 in \(F^{\infty }\) when \(a \to \infty \). Hence, by (4.1) we deduce that
This proves \(e^{ |\cdot |^{2}}\nu \in \Lambda _{0}\).
Conversely, assume that \(\{f_{n}\}\) is a bounded sequence in \(F^{p}\), and \(f_{n}\) uniformly converges to 0 on compact subsets of \(\mathbb{C}\) as \(n\to \infty \). It follows from Lemma 2.5 that
Since \(e^{ |\cdot |^{2}}\nu \in \Lambda _{0}\) by Lemma 2.7, \(\widetilde{(e^{ |\cdot |^{2}}\nu )}_{1}(z)\to 0\) as \(z \to \infty \). So, for any \(\varepsilon > 0\), there is some \(R>0\) such that when \(|z|\geq R\)
When \(|z|< R\), for the above ε, there is some \(R_{1}>0\) such that
Hence, by Lemma 2.5,
where the last inequality comes from the assumption that \(\{f_{n}\}\) converges to 0 uniformly on compact subsets of \(\mathbb{C}\) as \(n\to \infty \). Therefore, by the arbitrariness of ε and Lemma 4.1, we see that \(\mathcal{H}_{\nu}:F^{p} \to F^{\infty }\) is compact. □
Define the Rademacher functions \(\psi _{n}(t)\) on \([0,1]\) by
Then Khinchine’s inequality is the following, which can be found in [12].
Khinchine’s inequality
For \(0< p<\infty \) there exist constants \(0< a_{p}\leq B_{p}<\infty \) such that, for all natural numbers m and all complex numbers \(c_{1}, c_{2},\ldots,c_{m}\), we have
The next lemma is a partial result about atomic decomposition for Fock spaces, which can be found as Theorem 2.34 in [18].
Lemma 4.2
Let \(0< p\leq \infty \). For \(\lambda = \{\lambda _{j}\}^{\infty}_{j=1}\in l^{p}\), set
then S is a bounded operator from \(l^{p}\) to \(F^{p}\).
Theorem 4.3
Suppose \(0 < p <\infty \). Let ν be a nonnegative Borel measure on \([0, \infty )\) such that \(e^{\epsilon |\cdot |^{2}}\nu \in \Lambda \) with any fixed \(\epsilon >\frac{1}{2}\). Then the following statements are equivalent:
-
(i)
\(\mathcal{H}_{\nu}\) is a bounded operator from \(F^{\infty }\) into \(F^{p} \);
-
(ii)
\(\mathcal{H}_{\nu}\) is a compact operator from \(F^{\infty }\) into \(F^{p} \);
-
(iii)
\((\widetilde{e^{ |\cdot |^{2}}\nu})_{t}(z)\in L^{p}(dA)\) for some (or any) \(t>0\);
-
(iv)
\((\widehat{e^{ |\cdot |^{2}}\nu})_{r}(z)\in L^{p}(dA)\) for some (or any) \(r>0\);
-
(v)
The sequence \(\{(\widehat{e^{ |\cdot |^{2}}\nu})_{r}(a_{k})\}_{k=1}^{\infty}\in l^{p}\) for some (or any) r-lattice \(\{a_{k}\}_{k=1}^{\infty}\).
Proof
By [10, Lemma 2.3], we get the equivalence of (iii), (iv), and (v). The implication (ii) ⇒ (i) is trivial.
(i) ⇒ (v). Given any bounded sequence \(\{\lambda _{j}\}^{ \infty}_{j=1}\) and r-lattice \(\{a_{j}\}^{\infty}_{j=1}\), Lemma 4.2 shows that \(f(z) =\sum_{j=1}^{\infty}\lambda _{j} k_{a_{j}}(z) \in F^{\infty }\) with \(\|f\|_{\infty }\lesssim \|\{\lambda _{j}\}_{j}\|_{l^{\infty}}\). Note that \(\mathcal{H}_{\nu}: F^{\infty }\to F^{p} \) is bounded. By Khinchine’s inequality and Fubini’s theorem, we have
where \(\psi _{j}(t)\) is the jth Rademacher function on \([0,1]\). In addition, it follows from Lemma 2.4 and (4.1) that
Setting \(\beta _{k}=|\lambda _{k}|^{p}\), then \(\{\beta _{k}\}^{\infty}_{k=1}\in l^{\infty}\). Consequently,
The duality argument shows that \(\{\widehat{(e^{ |\cdot |^{2}}\nu )}_{r}(a_{k})^{p}\}^{\infty}_{k=1} \in l^{1}\), which means \(\{\widehat{(e^{ |\cdot |^{2}}\nu )}_{r}(a_{k})\}^{\infty}_{k=1} \in l^{p}\).
(iii) ⇒ (ii). Assume that \(\{f_{n}\}\) is a bounded sequence in \(F^{\infty}\), and \(f_{n}\) uniformly converges to 0 on compact subsets of \(\mathbb{C}\) as \(n\to \infty \). It follows from Lemma 2.5 that
Since \((\widetilde{e^{ |\cdot |^{2}}\nu})_{1}(z)\in L^{p}(dA)\), for any \(\varepsilon > 0\), there is some \(R>0\) such that
Note that \((\widetilde{e^{ |\cdot |^{2}}\nu})_{1}(z)\in L^{p}(dA)\) means \((\widehat{e^{ |\cdot |^{2}}\nu})_{r}(z)\in L^{p}(dA)\), hence \((\widehat{e^{ |\cdot |^{2}}\nu})_{r}(z)\in L^{\infty}(\mathbb{C})\). Lemma 2.6 yields that
Hence, for the above ε and R, there is some \(R_{1}>0\) such that
Together with the fact that \(f_{n}\) uniformly converges to 0 on compact subsets of \(\mathbb{C}\) as \(n\to \infty \), we obtain, by Lemma 2.5,
while n is large enough. Therefore, by the arbitrariness of ε and Lemma 4.1, we see that \(\mathcal{H}_{\nu}:F^{\infty }\to F^{p} \) is compact. □
The following result is a direct consequence of Theorem 4.3 and Lemma 2.8.
Corollary 4.1
Suppose \(1< p <\infty \). Let ν be a nonnegative Borel measure on \([0, \infty )\) that satisfies the condition in Theorem 3.1. Then the following statements are equivalent:
-
(i)
\(\mathcal{H}_{\nu}\) is a bounded operator from \(F^{\infty }\) into \(F^{p} \);
-
(ii)
\(\mathcal{H}_{\nu}\) is a compact operator from \(F^{\infty }\) into \(F^{p} \);
-
(iii)
\(e^{ |\cdot |^{2}}\nu \in \Lambda ^{\frac{p}{p-1}}\).
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Acknowledgements
The authors wish to thank the referees for their helpful comments and suggestions that greatly improved the paper.
Funding
This paper is supported by Guangdong Basic and Applied Basic Research Foundation (2020A1515110493 and 2020A1515111113), National Natural Science Foundation of China (12301093), and Natural Science Research Project of Guangdong Education Department (2020KQNCX042).
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Z. ZHUO, D. LI and T. ZENG developed the theoretical part and wrote this paper by themselves. T. ZENG helped perform the analysis with constructive discuss and read the manuscript. All authors reviewed the manuscript.
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Zhuo, Z., Li, D. & Zeng, T. A matrix acting between Fock spaces. J Inequal Appl 2024, 8 (2024). https://doi.org/10.1186/s13660-024-03084-7
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DOI: https://doi.org/10.1186/s13660-024-03084-7