Abstract
The space of entire functions \(\mathcal {F}_{\alpha }^{\infty }\) is mentioned in the paper of S. Janson, J. Peetre, and R. Rochberg. In this paper, we establish a characterization for the space \(\mathcal {F}_{\alpha }^{\infty }\) by \(n\)-th derivatives of entire functions. As an application of this result, we study the boundedness of Li-Stević’s integral operators and estimate essential norms of these operators acting on \(\mathcal {F}_{\alpha }^{\infty }\). Furthermore we describe complete characterizations for boundedness and compactness of the Volterra-type integral operators on \(\mathcal {F}_{\alpha }^{\infty }\).
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1 Introduction
Let \(dm\) denote the usual Lebesgue measure on the complex plane \({\mathbb {C}}\). For each \(p, 0 < p < \infty \hbox { and }\alpha > 0\), the Fock space \(\mathcal {F}_{\alpha }^p\) is the space of all entire functions \(f\) on \(\mathbb {C}\) for which
When \(p=2\), the space \(\mathcal {F}_{\alpha }^2\) is a functional Hilbert space with the inner product
The reproducing kernel function for \(\mathcal {F}_{\alpha }^2\) is given by \(K(z,w) = \exp \{\alpha z\overline{w}\}\) (see [6, Chapter 7]). We also use the normalized kernel function \(k_{w}(z) = \exp \{\alpha z\overline{w} -\frac{\alpha }{2}|w|^2\}\). Furthermore we note that \(\Vert k_{w}\Vert _{p, \alpha }^p = 1\) for each \(p > 0\). In Janson et al. [6], considered the space \(\mathcal {F}_{\alpha }^{\infty }\) which is defined by
In recent progresses of researches on the space \(\mathcal {F}_{\alpha }^{p}\), linear operators induced by entire functions have been studied by many authors. By Carswell et al. [1], boundedness and compactness of composition operators were characterized completely. Guo and Izuchi [5] investigate some operator theoretic properties of them. Choe et al. [2] study on a linear sum of two composition operators. Stroethoff [13] investigates the compactness of multiplication operator by using the Berezin symbol. For weighted composition operators acting on these spaces, we can refer to papers [11, 14, 15]. Recently Constantin [3] studies the Volterra-type operator on \(\mathcal {F}_{1}^p ~ (p >0)\). We consider a generalization of these type operators acting on \(\mathcal {F}_{\alpha }^{\infty }\).
Now let \(g\) and \(\varphi \) be entire functions on \(\mathbb {C}\). We define operators \(C_{\varphi }^{g}\hbox { and }T_{\varphi }^{g}\) by
for \(f\) is an entire function and \(z \in \mathbb {C}\). When \(g = {\varphi }^{\prime }\), the operator \(C_{\varphi }^{{\varphi }^{\prime }}\) is the composition operator \(C_{\varphi }\) up to a constant. If \(\varphi (z) = z\) and we replace \(g\) with \(g^{\prime }\), then \(T_{z}^{g^{\prime }}\) becomes the Volterra-type operator. The above operators \(C_{\varphi }^{g}\) and \(T_{\varphi }^{g}\) on some analytic function spaces over the unit disk of \(\mathbb {C}\) were introduced by Li and Stević [7, 8] and they investigated characterizations for boundedness and compactness of these operators. Lindström and Sanatpour [9] also considered boundedness and compactness of \(C_{\varphi }^{g}\) acting between Zygmund type spaces on the unit disk. Recently, Mengestie [10] studied the operator \(T_{\varphi }^{g^{\prime }} : \mathcal {F}_{\alpha }^{p} \rightarrow \mathcal {F}_{\alpha }^{q} \left( 0 < p, \, q < \infty \right) \).
Motivated by these studies, we will investigate the boundedness of \(C_{\varphi }^{g}\hbox { and }T_{\varphi }^{g}\) on \(\mathcal {F}_{\alpha }^{\infty }\). For the compactness of these operators, we give an estimate for the essential norm of them. To do our investigation, we need a characterization for the space \(\mathcal {F}_{\alpha }^{\infty }\) in terms of the first order derivative of entire functions. So our first purpose of this paper is to establish a characterization for \(\mathcal {F}_{\alpha }^{\infty }\) by higher order derivatives. By using this result, we will characterize the boundedness of \(C_{\varphi }^{g}\hbox { and }T_{\varphi }^{g}\), and estimate essential norms of them. Furthermore, as an application of our results, we describe a concrete form of \(g\) which induces the bounded or compact Volterra-type operators on \(\mathcal {F}_{\alpha }^{\infty }\).
Throughout this paper, the notation \(a {\lesssim }_{s,t} b\) means that there exists a positive constant \(C_{s,t}\) depending only on some parameters \(s, t\) such that \(a \le C_{s,t}b\). Moreover, if both \(a {\lesssim }_{s,t} b\hbox { and } b {\lesssim }_{s,t} a\) hold, then one says that \(a {\approx }_{s,t} b\).
2 Higher Order Derivatives Criterion for \(\mathcal {F}_{\alpha }^{\infty }\)
In this section, we shall obtain the \(n\)-th derivative criterion for \({\mathcal {F}}_{\alpha }^{\infty }.\) For the case of the usual Bloch space \(\mathcal {B}\) on the unit disk, a corresponding result have been obtained by Stroethoff [12].
Theorem 1
Suppose that \(n\) is a positive integer and \(f\) is an entire function. Then \(f \in \mathcal {F}_{\alpha }^{\infty }\) if and only if \(\frac{|f^{(n)}(z)|}{(1+|z|)^n}e^{-\frac{\alpha }{2}|z|^2} \in L^{\infty }(\mathbb {C})\).
Proof
First suppose that \(f \in \mathcal {F}_{\alpha }^{\infty }\). Now we consider the following operator:
By [6, Theorem 7.1], this \(P_{\alpha }\) is a bounded projection from \(\mathcal {L}_{\alpha }^{\infty }\) onto \(\mathcal {F}_{\alpha }^{\infty }\). Here \(\mathcal {L}_{\alpha }^{\infty }\) denotes the space of measurable functions \(g\) such that \(\mathrm{esssup}_{z \in \mathbb {C}}|g(z)|e^{-\frac{\alpha }{2}|z|^2} < \infty .\) Since \(\mathcal {F}_{\alpha }^{\infty } \subset \mathcal {L}_{\alpha }^{\infty }\), hence, we obtain
Differentiating under the integral sign gives
In the last line, we use an elementary inequality: \(1+|w| \le (1+|z|)(1+|w-z|)\) \((z, w \in \mathbb {C})\) which is verified by an application of the triangle inequality. Since
and so we obtain that
for all \(n \ge 1\).
To prove another direction, we may establish the following inequality:
for \(n \ge 1\). The above inequality is verified by induction on \(n\). First we consider the case \(n = 1\). For \(z \in \mathbb {C}\) we have that
The last integral in the above line is dominated by \(\frac{3}{\alpha }e^{\frac{\alpha }{2}|z|^2}\) when \(|z| > 1\) or \(2e^{\frac{\alpha }{2}|z|^2}\) when \(|z| \le 1\). Thus (3) gives that
and so (2) is true when \(n =1\). Next we assume that (2) holds when \(n = k\). By this assumption, we may prove that
As in (3), we have that
For \(z \in \mathbb {C}\) with \(|z| >1\),
On the other hand, we also obtain that
for \(z \in \mathbb {C}\) with \(|z| \le 1\). Hence we have that
for all \(z \in \mathbb {C}\), and so the desired estimate (4) is obtained. By induction on \(n\), we see that (2) is true for all \(n \ge 1\). \(\square \)
Remark
Inequality (1) also gives \(|f^{(j)}(0)| {\lesssim }_{\alpha , j} \Vert f\Vert _{\infty , \alpha }\) for all \(j \ge 1\). Combining this with (1) and (2), we obtain
In [6], Janson et al. have also mentioned the subspace \(\mathcal {F}_{\alpha ,0}^{\infty }\) of \(\mathcal {F}_{\alpha }^{\infty }.\) Namely, \(\mathcal {F}_{\alpha ,0}^{\infty }\) consists of all entire functions \(f\) which satisfy \(|f(z)|e^{-\frac{\alpha }{2}|z|^2} \rightarrow 0\) as \(|z| \rightarrow \infty \). It is expected that the \(n\)-th derivative criterion in Theorem 1 carries over to the space \(\mathcal {F}_{\alpha ,0}^{\infty }\).
For each positive integer \(n\), we denote
Now we will show that \(\mathcal {F}_{\alpha ,0}^{\infty }\) is equal to \(\mathcal {F}^{(n)}\).
Lemma 1
If \(f \in \mathcal {F}^{(n)}\), then \(\lim _{r \rightarrow 1^{-}}\Vert f-f_r\Vert _{\infty , \alpha } = 0\) where \(f_r(z) = f(rz)\) for \(0 \le r <1\).
Proof
Suppose that \(f \in \mathcal {F}^{(n)}\) and take \(\epsilon >0\) arbitrarily. Then there exists \(R>0\) such that
If \(1/2 < r < 1\) and \(|z| > 2R\), then we have that
The uniform continuity of \(f^{(n)}\) on the compact set \(\{z \, : \, |z| \le 2R\}\) implies that
and so we have that
for \(|z| \le 2R\) and \(r\) is close to \(1\) sufficiently. Combining these estimates with (5), we obtain that
By letting \(r \rightarrow 1^{-}\), we can get the result. \(\square \)
Proposition 1
\(\mathcal {F}^{(n)}\) is the closure of the polynomials in \(\mathcal {F}_{\alpha }^{\infty }\) under the norm \(\Vert \cdot \Vert _{\infty , \alpha }.\)
Proof
For \(f (z) = \sum _{j= 0}^{\infty }a_j z^j, 0<r<1\) and each positive integer \(N\), we put \(P_N (z) = \sum _{j= 0}^{N}a_j r^j z^j\). Cauchy’s integral formula implies that
Thus if we put \(t = \sqrt{\frac{j}{\alpha }}\), then we have that
Since \(\Vert z^j\Vert _{\infty , \alpha } = \left( \frac{j}{\alpha e}\right) ^{\frac{j}{2}}\), we see that
Now we assume that \(f \in \mathcal {F}^{(n)}\) and \(\epsilon >0\). By Lemma 1, there exists \(0< r_0 < 1\) such that \(\Vert f - f_{r_0}\Vert _{\infty , \alpha } < \epsilon \). By using the above argument, we also have that
Since \(f \in \mathcal {F}_{\alpha }^{\infty }\) by Theorem 1, this implies that \( \limsup _{N \rightarrow \infty }\Vert f- P_N\Vert _{\infty , \alpha } \le \epsilon \). This completes the proof. \(\square \)
Corollary 1
For each positive integer \(n\), an entire function \(f\) belongs to \(\mathcal {F}_{\alpha , 0}^{\infty }\) if and only if \(\frac{|f^{(n)}(z)|}{(1+|z|)^n}e^{-\frac{\alpha }{2}|z|^2} \rightarrow 0\) as \(|z| \rightarrow \infty \).
Proof
By noting that the polynomial set is dense in \(\mathcal {F}_{\alpha }^1\) (see [4, Proposition 5]) and \(\mathcal {F}_{\alpha }^{1}\) is dense in \(\mathcal {F}_{\alpha , 0}^{\infty }\) (see [6, Theorem 7.2]), we see that \(\mathcal {F}_{\alpha , 0}^{\infty }\) is also the closure of the polynomials in \(\mathcal {F}_{\alpha }^{\infty }\) under the norm \(\Vert \cdot \Vert _{\infty , \alpha }\). Hence Proposition 1 implies \(\mathcal {F}_{\alpha , 0}^{\infty } = \mathcal {F}^{(n)}\). We obtain the desired characterization for the space \(\mathcal {F}_{\alpha , 0}^{\infty }\). \(\square \)
3 Applications to Operators on \(\mathcal {F}_{\alpha }^{\infty }\)
In this section, we investigate linear operators induced by entire functions \(g\) and \(\varphi \). First we consider the boundedness of \(C_{\varphi }^{g}\) on \(\mathcal {F}_{\alpha }^{\infty }\) and the estimate for its essential norm.
Theorem 2
The operator \(C_{\varphi }^{g}\) is bounded on \(\mathcal {F}_{\alpha }^{\infty }\) if and only if \(\varphi \) and \(g\) satisfy
If \(C_{\varphi }^{g}\) is bounded on \(\mathcal {F}_{\alpha }^{\infty }\), then it holds that
Here \(\Vert C_{\varphi }^{g}\Vert _{e}\) denotes the essential norm of \(C_{\varphi }^{g}\). Hence \(C_{\varphi }^{g}\) is compact if and only if
Proof
By applying Theorem 1 to \(\Vert C_{\varphi }^{g}f\Vert _{\infty , \alpha }\), we see that condition (6) is a sufficient condition for the boundedness of \(C_{\varphi }^{g}\).
To prove the other direction, we consider the function \( f_{z}(w) = k_{\varphi (z)}(w) \) for fixed \(z \in \mathbb {C}\) with \(|\varphi (z)|>1\). Since \(f_z \in \mathcal {F}_{\alpha }^{\infty }\) and \(\Vert f_z\Vert _{\infty , \alpha } \le 1\), it follows from Theorem 1 that
By considering the identity function \(\pi (z) = z\), we see that the boundedness of \(C_{\varphi }^{g}\) implies that
and so we have that
for any \(z \in \mathbb {C}\) with \(|\varphi (z)| \le 1\). Hence (6) is verified by combining this with (8) and (9).
Next we will prove the estimate (7). To prove the lower estimate in (7), we take a sequence \(\{z_j\}\) in \(\mathbb {C}\) with \(|\varphi (z_j)| \rightarrow \infty \) as \(j \rightarrow \infty \). Put \(f_j(z) = k_{\varphi (z_j)}(z)\). Then \(\{f_j\}\) is a bounded sequence in \(\mathcal {F}_{\alpha }^{\infty }\) and \(f_j \rightarrow 0 ~ (\text {as} \, j \rightarrow \infty )\) uniformly on compact subsets in \(\mathbb {C}\). More precisely, we see that \(f_j \in \mathcal {F}_{\alpha , 0}^{\infty }\) for each \(j\). By using the duality \((\mathcal {F}_{\alpha , 0}^{\infty })^{*} \cong \mathcal {F}_{\alpha }^1\) (see [6, Theorem 7.4]), a standard argument shows that \(\{f_j\}\) converges to \(0\) weakly in \(\mathcal {F}_{\alpha , 0}^{\infty }\). Since \((\mathcal {F}_{\alpha }^{\infty })^{*} \subset (\mathcal {F}_{\alpha , 0}^{\infty })^{*}\), this shows that \(\{f_j\}\) also converges to \(0\) weakly in \(\mathcal {F}_{\alpha }^{\infty }\). Hence we see that \(\Vert \mathcal {K}f_{j}\Vert _{\infty , \alpha } \rightarrow 0\) as \(j \rightarrow \infty \) for any compact operators \(\mathcal {K}\) on \(\mathcal {F}_{\alpha }^{\infty }\) and
As in the argument of (8), we also obtain that
for a sufficiently large \(j\). Thus we get
and so this implies the desired lower estimate for \(\Vert C_{\varphi }^{g}\Vert _{e}\).
Now we prove the upper estimate in (7). For each positive integer \(k\) we put \({\phi }_{k}(z) = \frac{k}{k+1}z.\) Then it follows from [15, Theorem 4 (b)] (also see [1]) that the composition operator \(C_{{\phi }_k}\) is compact on \(\mathcal {F}_{\alpha }^{\infty }\). Hence we have that
where \(I\) denotes the identity operator on \(\mathcal {F}_{\alpha }^{\infty }\). By Theorem 1 the last term in (10) is comparable with
for any fixed \(r \in (0, \infty )\). By an application of Theorem 1 once again, we see that the second term in (11) is dominated by
Note that this estimate does not depend on \(k\).
Next assertion is
Put \(w = \varphi (z)\) and denote \(\left[ \frac{k}{k+1}w, w\right] \) the radial segment. By integrating \(f^{\prime }\) along \(\left[ \frac{k}{k+1}w, w\right] \), we obtain that
for some \(\xi (w) \in \left[ \frac{k}{k+1}w, w\right] \). Furthermore an application of Cauchy’s estimate to \(f^{\prime \prime }\) on the circle with center at \(\xi (w)\) and radius \(r\) gives that
Hence we have that
for fixed \(r \in (0, \infty )\). Combinig this with (9) prove (13). By (10)–(13), we obtain that
Since \(r \in (0, \infty )\) was arbitrary, we get the desired upper estimate in (7). \(\square \)
For the operator \(T_{\varphi }^{g}\), on the other hand, Theorem 1 gives that
and so the condition
is a sufficient condition for the boundedness of \(T_{\varphi }^{g}\) on \(\mathcal {F}_{\alpha }^{\infty }\). Furthermore the test function \(k_{\varphi (z)}(w)\) implies that this condition is also a necessary condition. By the same argument in Theorem 2, we can obtain an estimate for the essential norm \(\Vert T_{\varphi }^{g}\Vert _{e}\). Hence we obtain the following result.
Theorem 3
The operator \(T_{\varphi }^{g}\) is bounded on \(\mathcal {F}_{\alpha }^{\infty }\) if and only if \(\varphi \) and \(g\) satisfy
If \(T_{\varphi }^{g}\) is bounded on \(\mathcal {F}_{\alpha }^{\infty }\), then the essential norm \(\Vert T_{\varphi }^{g}\Vert _{e}\) is comparable with
For each entire functions \(g\), integral type operators \(J_{g}\) and \(I_{g}\) are defined by
respectively. These operators are the special cases of \(T_{\varphi }^{g}\) and \(C_{\varphi }^{g}\). In fact, if we replace \(g\) with \(g^{\prime }\) and put \(\varphi (z) = z\) in the definition of \(T_{\varphi }^{g}\), then we have that \(T_{z}^{g^{\prime }}f = J_{g}f\). On the other hand, by the definition of \(C_{\varphi }^{g}\), we see that \(C_{z}^{g}f = I_{g}f\). Hence we can describe the characterizations for the boundedness and compactness of \(J_{g}\) and \(I_{g}\) as an application of Theorems 2 and 3.
Corollary 2
For each entire function \(g\), the following hold.
- (a):
-
\(J_{g}\) is bounded on \(\mathcal {F}_{\alpha }^{\infty }\) if and only if \(g\) is a polynomial of degree \(\le 2\).
- (b):
-
\(J_{g}\) is compact on \(\mathcal {F}_{\alpha }^{\infty }\) if and only if \(g\) is a polynomial of degree \(\le 1\).
- (c):
-
\(I_{g}\) is bounded on \(\mathcal {F}_{\alpha }^{\infty }\) if and only if \(g\) is a constant function.
- (d):
-
\(I_{g}\) is compact on \(\mathcal {F}_{\alpha }^{\infty }\) if and only if \(g \equiv 0\).
Proof
Claims (c) and (d) are an immediate consequence of Theorem 2. So we may only prove (a) and (b). By Theorem 3 we see that \(J_{g}\) is bounded if and only if \(g\) satisfies
This implies that \(g^{\prime }\) must be a polynomial of degree \(\le 1\), and so the degree of \(g\) is less than or equal to \(2\). Theorem 3 also says that
is equivalent to the compactness of \(J_{g}\), and so \(g^{\prime }\) is constant. Hence \(J_{g}\) is compact if and only if the degree of \(g\) must be less that or equal to \(1\). \(\square \)
Remark
The claim (a) and (b) for the Fock space \(\mathcal {F}^p (= \mathcal {F}_{1}^p) ~ (0<p< \infty )\) is proved by Constantin [3, Theorem 1].
Finally we will mention the action of differential and integral operators on \(\mathcal {F}_{\alpha }^{\infty }\). We put
for each entire function \(f\). Take a sequence \(\{z_j\}\) in \(\mathbb {C}\) with \(|z_j| \rightarrow \infty \) as \(j \rightarrow \infty \) and \(f \in \mathcal {F}_{\alpha }^{\infty }\). Then we have
By applying Theorem 1 to the above, we see that \(D\) is not bounded on \(\mathcal {F}_{\alpha }^{\infty }\). Since \(If(z) = J_{z}f(z)\), on the other hand, Corollary 2 implies that \(I\) is always bounded (and compact) on \(\mathcal {F}_{\alpha }^{\infty }\).
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Acknowledgments
The author thanks S. Ohno and E. Saukko for valuable discussions about the material in the paper. Especially, I could improve the proof of Theorem 1 by Saukko’s suggestions. The author is partly supported by the Grants-in-Aid for Young Scientists (B, No. 23740100), Japan Society for the Promotion of Science (JSPS).
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Communicated by Aurelian Gheondea.
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Ueki, SI. Characterization for Fock-Type Space via Higher Order Derivatives and its Application. Complex Anal. Oper. Theory 8, 1475–1486 (2014). https://doi.org/10.1007/s11785-013-0333-3
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DOI: https://doi.org/10.1007/s11785-013-0333-3