Abstract
Let \(B_E\) be the open unit ball of a complex finite- or infinite-dimensional Hilbert space. If f belongs to the space \(\mathcal {B}(B_E)\) of Bloch functions on \(B_E\), we prove that the dilation map given by \(x \mapsto (1-\Vert x\Vert ^2) \mathcal {R}f(x)\) for \(x \in B_E\), where \(\mathcal {R}f\) denotes the radial derivative of f, is Lipschitz continuous with respect to the pseudohyperbolic distance \(\rho _E\) in \(B_E\), which extends to the finite- and infinite-dimensional setting the result given for the classical Bloch space \(\mathcal {B}\). To provide this result, we will need to prove that \(\rho _E(zx,zy) \le |z| \rho _E(x,y)\) for \(x,y \in B_E\) under some conditions on \(z \in \mathbb {C}\). Lipschitz continuity of \(x \mapsto (1-\Vert x\Vert ^2) \mathcal {R}f(x)\) will yield some applications on interpolating sequences for \(\mathcal {B}(B_E)\) which also extends classical results from \(\mathcal {B}\) to \(\mathcal {B}(B_E)\). Indeed, we show that it is necessary for a sequence in \(B_E\) to be separated to be interpolating for \(\mathcal {B}(B_E)\) and we also prove that any interpolating sequence for \(\mathcal {B}(B_E)\) can be slightly perturbed and it remains interpolating.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction and background
Let \(\textbf{D}\) be the open unit disk of the complex plane \(\mathbb {C}\) and \(\rho \) the pseudohyperbolic distance defined by \(\rho (z,w)=\left| \frac{z-w}{1-z {\bar{w}}} \right| \). Consider a function f of the classical Bloch space \(\mathcal {B}\). The study of the Lipschitz continuity of the dilation map \(z \mapsto (1-|z|^2)f'(z)\) with respect to \(\rho \) was first started by Attele to study conditions for a sequence on \(\textbf{D}\) to be interpolating for \(\mathcal {B}\) (see [2]). Ghatage, Yan, and Zheng also discussed a similar result providing a different proof [8]. Then, Xiong improved this result in [14] and other authors extended it for high-dimensional holomorphic functions [6] and planar harmonic mappings [7]. For bounded symmetric domains of \(\mathbb {C}^n\), see also [10].
The aim of this paper is to extend the previous result for the infinite-dimensional setting and give some applications. Other applications on bounded below composition operators can be found in [12]. Along this work, E will denote a finite- or infinite-dimensional complex Hilbert space and its open unit ball will be denoted by \(B_E\). In Sect. 2, we will study the boundness of
for \(z \in \mathbb {C}\) and \(x,y \in B_E\), such that \(zx,zy \in B_E\) proving that, in general, this expression is unbounded. Nevertheless, we will show that if |z| is bounded above by
then the expression above is bounded by 2 and this bound is the best possible. We first prove the case when we deal with \(E=\mathbb {C}\) and then with any finite- or infinite-dimensional Hilbert space E. At the end of this section, we will extend this result to the case when we deal with the Banach space \(C_0(S)\).
In Sect. 3, we will consider the space \(\mathcal {B}(B_E)\) of Bloch functions f on \(B_E\). Recall that for \(f \in \mathcal {B}(B_E)\) and \(x \in B_E\), the radial derivative Rf(x) is given by \(Rf(x)=\langle x,f'(x) \rangle .\) As a consequence of the boundness of (1.1), we show that the dilation map \(x \mapsto (1-\Vert x\Vert ^2) \mathcal {R}f(x)\) for \(x \in B_E\) is Lipschitz continuous with respect to the pseudohyperbolic distance \(\rho _E\) in Sect. 3.1, extending to the finite- and infinite-dimensional setting the results mentioned above. Hence, we derive some results about interpolating sequences for \(\mathcal {B}(B_E)\) in Sect. 3.2. Indeed, we supply a proof that these sequences are separated for the pseudohyperbolic distance. We also prove that interpolating sequences can be slightly perturbed and they remain interpolating, which also extends the result for \(\mathcal {B}\) given in [2].
1.1 The pseudohyperbolic and hyperbolic distance
Let \(\textbf{D}\) be the open unit disk of the complex plane \(\mathbb {C}\). As we have mentioned, the pseudohyperbolic distance for \(z,w \in \textbf{D}\) is given by
If we deal with a complex Banach space X with open unit ball \(B_X\), recall that \(f: B_X \rightarrow \mathbb {C}\) is said to be holomorphic (or analytic) if it is Fréchet differentiable for any \(x \in B_X\) (see [12] for further information). For any \(x,y \in B_X\), the pseudohyperbolic distance \(\rho _X(x,y)\) is given by
where \(H^{\infty }(B_X)\) is the space of bounded holomorphic functions on \(B_X\) which become a Banach space (a uniform algebra, indeed) endowed with the sup-norm. The hyperbolic distance for \(x,y \in B_X\) is given by
1.2 Automorphisms and pseudohyperbolic distance on \(B_E\)
If we deal with a complex Hilbert space E, we will denote by \(\textrm{Aut}(B_E)\) the space of automorphisms of \(B_E\), that is, the maps \(\varphi : B_E \rightarrow B_E\) which are bijective and bianalytic (see [9]). For any \(x \in B_E\), the automorphism \(\varphi _x: B_{E} \longrightarrow B_{E}\) is defined according to
where \(s_x=\sqrt{1-\Vert x\Vert ^2},\) \(m_x: B_{E} \longrightarrow B_{E}\) is the analytic self-map
\(P_x: E \longrightarrow E\) is the orthogonal projection along the one-dimensional subspace spanned by x, that is
and \(Q_x: E \longrightarrow E\) is the orthogonal complement \(Q_x=\textrm{Id}_E-P_x\), where \(\textrm{Id}_E\) denotes the identity operator on E. It is clear that \(\varphi _x(0)=x\) and \(\varphi _x(x)=0\). The automorphisms of the unit ball \(B_{E}\) turn to be compositions of these \(\varphi _x\) with unitary transformations U of E.
It is well known (see [9]) that the pseudohyperbolic distance on \(B_E\) is given by
and
1.3 The Bloch space
The classical Bloch space \(\mathcal {B}\) is the set of holomorphic functions \(f: \textbf{D}\rightarrow \mathbb {C}\), such that \(\Vert f\Vert _B=\sup _{z \in \textbf{D}} (1-|z|^2)|f'(z)|\) is bounded. This supremum defines a semi-norm which becomes a norm by adding up a constant: \(|f(0)|+\sup _{z \in \textbf{D}} (1-|z|^2)|f'(z)|\). Hence, \(\mathcal {B}\) becomes a complex Banach space. The semi-norm \(\Vert \cdot \Vert _B\) is invariant by automorphisms, that is, \(\Vert f \circ \varphi \Vert _B=\Vert f\Vert _B\) for any \(f \in \mathcal {B}\) and \(\varphi : \textbf{D}\rightarrow \textbf{D}\) an automorphism of \(\textbf{D}\).
Timoney extended Bloch functions if we deal with a finite-dimensional Hilbert space (see [13]). Blasco, Galindo, and Miralles extended them to the infinite-dimensional setting (see [5]). If we deal with a complex finite- or infinite-dimensional Hilbert space E, the analytic function \(f: B_E \rightarrow \mathbb {C}\) is said to belong to the Bloch space \(\mathcal {B}(B_E)\) if
where it is clear that \(\nabla f(x)\) is the derivative \(f'(x)\) or, equivalently, if
where \(\mathcal {R}f(x)\) is the radial derivative of f at x given by \(\mathcal {R}f(x)=\langle x, \overline{\nabla f (x)}\rangle \). These semi-norms are equivalent to the following one:
where \({\widetilde{\nabla }} f (x)\) denotes the invariant gradient of f at x which is given by \({\widetilde{\nabla }} f (x) =\nabla (f \circ \varphi _x)(0)\), where \(\varphi _x\) is the automorphism given in (1.2).
The three semi-norms \(\Vert \cdot \Vert _\mathcal {B}, \Vert \cdot \Vert _\mathcal {R}\) and \(\Vert \cdot \Vert _\mathcal {I}\) define equivalent Banach space norms-modulo the constant functions- in \(\mathcal {B}(B_E)\) (see [5]). In particular, there exists a constant \(A_0 >0\), such that
Hence, the space \(\mathcal {B}(B_E)\) can be endowed with any of the norms \(\Vert \cdot \Vert _{\mathcal {B}-Bloch}=|f(0)|+ \Vert \cdot \Vert _\mathcal {B}\) or \(\Vert \cdot \Vert _{\mathcal {R}-Bloch}=|f(0)|+ \Vert \cdot \Vert _\mathcal {R}\) or \(\Vert \cdot \Vert _{\mathcal {I}-Bloch}=|f(0)|+ \Vert \cdot \Vert _\mathcal {I}\) and \(\mathcal {B}(B_E)\) becomes a Banach space. We will make use of these three semi-norms and norms along the sequel. We will also make use of this result, which states that Bloch functions on \(B_E\) are Lipschitz with respect to the hyperbolic distance (see [3]):
Proposition 1.1
Let E be a complex Hilbert space and let \(f \in \mathcal {B}(B_E)\). Then, for any \(x,y \in B_E\)
2 Inequalities with the pseudohyperbolic distance
Let X be a complex Banach space. If \(\varphi : B_X \rightarrow B_X\) is an analytic self-map, it is well known that \(\rho _X(\varphi (x),\varphi (y)) \le \rho _X (x,y)\) for any \(x,y \in B_X\) and the equality is attained if and only if \(\varphi \) is an automorphism of \(B_X\). Hence, if we consider \(z \in \mathbb {C}\), \(|z| \le 1\) and \(x,y \in B_X\), it is clear that
since the map \(\varphi : B_X \rightarrow B_X\) given by \(\varphi (x)=z x\) is analytic on \(B_X\). However, this situation changes dramatically if we consider the expression
for any \(z \in \mathbb {C}\), such that \(zx,zy \in B_X\). We show that, in general, this expression is unbounded. Anyway, if we deal with X a complex Hilbert space or \(C_0(S)\) and \(z \in \mathbb {C}\) satisfies
then expression (2.1) is bounded by 2 and this will permit us to provide several applications in Sect. 3.
2.1 Unboundness
In this section, we prove that expression (2.1) is unbounded in general.
Proposition 2.1
Let E be a complex Hilbert space. There exist a sequence \((z_n) \subset \mathbb {C}\), \(x \in B_E\) and a sequence \((y_n) \subset B_E\), such that \(z_n x,z_n y_n \in B_E\) but
is unbounded.
Proof
We prove it for \(E=\mathbb {C}\). Take for instance \(x=1/2\), \(y_n=1/2-\frac{1}{n}\) and \(z_n =2-\frac{1}{n}\). It is clear that \(|z_n x|<1\) and \(|z_n y_n| <1\). However
and
Hence
which is clearly unbounded, since it tends to \(\infty \) when \(n \rightarrow \infty \). The result remains true if we deal with any complex Hilbert space E, since we can take \(x_0 \in E\), such that \(\Vert x_0\Vert =1\) and take \(u=\frac{1}{2} x_0\), \(v_n= y_n x_0\). We have that
and similarly, we have \(\rho _E(u,v_n)=\rho _E(x,y_n)\), so we apply the case \(E=\mathbb {C}\) and we are done. \(\square \)
An easy consequence is a well-known result: the pseudohyperbolic distance cannot be extended to a norm on E, since
is unbounded, so \(\rho _E(zx,zy) \ne |z| \rho _E(x,y)\).
2.2 Boundness
The main result of this section is Theorem 2.7 which states that under condition (2.1), then the expression (1.1) is bounded and the best bound possible is given by 2. The following lemma will be used to prove this result.
Lemma 2.2
Let E be a finite- or infinite-dimensional Hilbert space, \(z \in \mathbb {C}\) and \(x,y \in B_E\), such that
Then, \(|1-p| \le 2 |1-|z|^2 p|\) where p denotes the scalar product \(\langle x,y \rangle \).
Proof
Suppose without loss of generality that \(\Vert x\Vert \ge \Vert y\Vert \) and \(x \ne 0\). Otherwise, the inequality is clearly true for any \(z \in \mathbb {C}\). Notice that
so it is sufficient to prove that \(|1-|z|^2p|+||z|^2-1||p| \le 2|1-|z|^2p|\), which is equivalent to \(|1-|z|^2p| \ge ||z|^2-1||p|\). We consider two cases:
i) if \(|z|^2 \le 1\), then \(1-|z|^2 \ge 0\), so we need to prove \(|1-|z|^2 p| \ge (1-|z|^2)|p|\) which is clearly satisfied, since
where second inequality is true, because \(|p| \le \Vert x\Vert \Vert y\Vert < 1\).
ii) On the other hand, suppose that \(|z|^2 >1\): we need to prove that \(|1-|z|^2p| \ge (|z|^2-1)|p|\). Since \(|1-|z|^2p| \ge 1-|z|^2 |p|\), it is sufficient to prove that
Notice that \(1-|z|^2|p| >0\), since \(|z|^2 |p| \le \Vert zx\Vert \Vert zy\Vert < 1\), because \(zx,zy \in B_E\). Inequality (2.2) is equivalent to \(2|z|^2 |p| <1+|p|\) which is true, since
so we need to prove that
However
where last inequality is true because of the arithmetic mean-geometric mean inequality and since \(|p| \le \Vert x\Vert \Vert y\Vert \le \Vert x\Vert ^2\). \(\square \)
2.2.1 The case \(E=\mathbb {C}\)
If we deal with \(E=\mathbb {C}\), it is easy to prove that (2.1) is bounded:
Proposition 2.3
Let \(x,y \in \textbf{D}\) and \(z \in \mathbb {C}\), such that
Then, \(zx,zy \in \textbf{D}\) and
Proof
Suppose without loss of generality that \(|x| \ge |y|\) and take \(z \ne 0\) (otherwise, it is clear). Notice that \(zx,zy \in \textbf{D}\), since
We have
and
so the inequality is equivalent to
Calling \(p=x{\overline{y}}\), we have to prove that \(|1-p| \le 2 |1-|z|^2 p|\). Apply Lemma 2.2 for \(E=\mathbb {C}\) and we are done. \(\square \)
Remark 2.4
Notice that the bound 2 is the best possible. Indeed, take \(x_n,y_n \in \textbf{D}\), such that \(x_n \rightarrow 1\) and \(y_n \rightarrow -1\). It is clear that for \(z_n \rightarrow 0\), the expression \(|1-\overline{x_n}y_n|\) tends to 2 when \(n \rightarrow \infty \) and the expression \(2 |1-|z_n|^2\overline{x_n}y_n|\) also tends to 2, so the inequality above is sharp. \(\square \)
2.2.2 The case when E is any complex Hilbert space
We will deal with \(x,y \in \mathcal {B}_E\) and \(z \in \mathbb {C}\), such that \(zx,zy \in B_E\), we will denote by \(r=\Vert x\Vert \), \(s=\Vert y\Vert \) and \(u=\Vert x\Vert ^2+\Vert y\Vert ^2=r^2+s^2\). We will also denote by p the scalar product \(\langle x,y \rangle \) and \(m=\Re p\). This notation will be used in Lemma 2.5 and Theorem 2.7.
Lemma 2.5
Let E be a complex Hilbert space and \(x,y \in B_E\). Then
Proof
The inequality is equivalent to
Bearing in mind that \(m=\Re p\) and \(u=r^2+s^2\), we need to prove
Notice that \(|p| \ge |m|\) so \((4+2\,m-u) |p|^2 \ge ~(4+m-u) m^2\) since \(4+2\,m-u \ge 0\). Hence, it is sufficient to prove
It is also clear that \(u^2 \ge 4r^2s^2\), so last inequality is equivalent to
The expression at left can be easily factorized and is equal to
where both factors are clearly greater or equal to 0 and we are done. \(\square \)
This lemma will be used at the end of the proof of Theorem 2.7:
Lemma 2.6
Let \(f(a,b,c)=(3-b^2)(a-c)-(a^2-b^2)(2-c)\). Then, \(f(a,b,c) \ge 0\) for any \(0 \le c \le b \le a \le 1\).
Proof
Notice that \(f(a,b,c)=(3-b^2)a-2(a^2-b^2)-(3-a^2)c\), so f is affine with respect to c. Hence, it is enough to prove the inequality for \(c=b\). The function becomes \(f(a,b,b)=(3-b^2)(a-b)-(a^2-b^2)(2-b)=(a-b)((3-b^2)-(a+b)(2-b))\), and since \(a-b \ge 0\), it is enough to prove that \((3-b^2)-(a+b)(2-b) \ge 0\). The expression \(g(a,b)=(3-b^2)-(a+b)(2-b)\) is affine with respect to a, so it is enough to prove it for \(a=1\). Notice that \(g(1,b)=(3-b^2)-(1+b)(2-b)=1-b\) which is clearly greater or equal to 0, so we are done. \(\square \)
Theorem 2.7
Let E be a finite- or infinite-dimensional complex Hilbert space, \(z \in \mathbb {C}\) and \(x,y \in B_E\). If
then \(zx,zy \in B_E\) and
Proof
Suppose without loss of generality that \(\Vert x\Vert \ge \Vert y\Vert \) and \(z \ne 0\). We will denote \(\rho =\rho _E(x,y)\) and \(\rho _z=\rho _E(zx,zy)\). If \( \frac{1}{2} \le |z| < 1\), then the result is clear, since
where first inequality is true because of the contractivity of the pseudohyperbolic distance for the function \(g: B_E \rightarrow B_E\) given by \(g(x)=z x\).
Therefore, let us prove it for \(|z| < 1/2\) or \(|z| \ge 1\). Taking squares, the inequality is equivalent to prove
Bear in mind the expression (1.4) for the pseudohyperbolic distance and call \(t=|z|^2\) which is different from 0, since \(z \ne 0\). We have
We will introduce the following notation:
Notice that \(A-B \ge 0\), since
Using this notation, inequality (2.3) is equivalent to
so bearing in mind that \(t=|z|^2\), we only need to prove (2.6) for \(t \ge 1\) and \(t \le 1/4\). If \(t \ge 1\), the result is clear, since
where last inequality is true by Lemma 2.2. Therefore, it remains to prove it for \(0 \le t \le 1/4\). Inequality 2.6 is equivalent to
Since \(B,t \ge 0\) and \(4(A-B)-A|1-p|^2 \ge 0\) by Lemma 2.5, this inequality is clearly true if \(m <0\). Therefore, we can suppose without loss of generality that \(m \ge 0\). The inequality is equivalent to
so we will prove last inequality. Notice that
Since \(0 \le t \le 1/4\), we have that \(3/4 \le 1-t \le 1\), so
Bearing in mind (2.4) and (2.5), notice that
where f is the function defined in Lemma 2.6 and since \(0 \le m \le |p| \le rs \le 1\), using the lemma, we are done, since
\(\square \)
2.2.3 Results for \(X=C_0(S)\)
Let S be a locally compact topological space and consider \(X=C_0(S)\) given by the space of continuous functions \(f: S \rightarrow \mathbb {C}\), such that for any \(\varepsilon >0\), there exists a closed compact subset \(K \subset S\), such that \(|f(x)| < \varepsilon \) for any \(x \in S {\setminus } K\). Endowed with the sup-norm, \(C_0(S)\) becomes a Banach space and the pseudohyperbolic distance for \(x,y \in C_0(S)\) is well known (see [1]) and it is given by
We prove that expression (2.1) is also bounded by 2 when we deal with the space \(X=C_0(S)\):
Proposition 2.8
Let \(X=C_0(S)\) and \(x,y \in X\). If \(z \in \mathbb {C}\) satisfies
then
Proof
Suppose without loss of generality that \(\Vert x\Vert \ge \Vert y\Vert \). For any \(t \in S\), we have that \(x(t),y(t) \in \textbf{D}\), since \(\Vert x\Vert =\sup _{t \in S}|x(t)| <1\) and \(\Vert y\Vert =\sup _{t \in S} |y(t)|<1\). The result is clear, since
where first inequality is clear because of Proposition 2.3 and because for any \(t \in X\), we have that
and we are done. \(\square \)
3 Applications
Theorem 2.7 yields several applications. As we have mentioned, we first show that for a Bloch function \(f: B_E \rightarrow \mathbb {C}\), the function \(x \mapsto (1-\Vert x\Vert ^2) |\mathcal {R}f(x)|\) for \(x \in B_E\) is Lipschitz continuous with respect to the pseudohyperbolic distance \(\rho _E\). Hence, we derive some results about interpolating sequences for \(\mathcal {B}(B_E)\) in Sect. 3.2. Indeed, we provide a new proof that these sequences are separated for the pseudohyperbolic distance. We also prove that these sequences can be slightly perturbed and they remain interpolating.
3.1 The Lipschitz continuity of \((1-\Vert x\Vert ^2) |\mathcal {R}f(x)|\)
We will denote by \(\Pi \) the unit circle of the complex plane \(\mathbb {C}\), that is, the set of complex numbers u, such that \(|u|=1\).
Lemma 3.1
Let \(f \in \mathcal {B}(B_E)\). Fix \(\varepsilon >0\) and \(x,y \in B_E\). If \((1+\varepsilon u)x\) and \((1+\varepsilon u)y\) belongs to \(B_E\) for any \(u \in \Pi \), then there exists \(u_0 \in \Pi \), such that
Proof
Fix \(x,y \in B_E\) and \(\varepsilon >0\). Notice that the function \(f(x+\varepsilon u x)-f(y+\varepsilon u y)\) defined for \(u \in \Pi \) is continuous. Since \(\Pi \) is a compact set, there exists \(u_0 \in \Pi \), such that
Consider \(g(u)=f(x+\varepsilon u x)\) for u defined on an open disk of the complex plane \(\mathbb {C}\) which contains \(\Pi \). It is clear that
so \(g'(0)=\varepsilon \mathcal {R}f(x)\). Similarly, if \(h(u)=f(y+\varepsilon u y)\), then \(h'(0)=\varepsilon R f(y)\). By the Cauchy’s integral formula, we have
where last inequality is true by Proposition 1.1. \(\square \)
The proof of the following lemma is an easy calculation. It will be used in Lemma 3.3.
Lemma 3.2
For any \(0 \le t < 1\), we have
Lemma 3.3
Let \(f \in \mathcal {B}(B_E)\) and \(x,y \in B_E\), such that \(\Vert x\Vert \ge \Vert y\Vert \). Then
Proof
Take
Notice that for any \(u \in \Pi \), we have that \((1+\varepsilon u)x\) and \((1+\varepsilon u) y\) belongs to \(B_E\), since
so clearly \(\Vert (1+\varepsilon u) x\Vert \le (1+\varepsilon ) \Vert x\Vert < 1\) and since \(\Vert y\Vert \le \Vert x\Vert \)
By Lemma 3.1, there exists \(u_0 \in \Pi \), such that:
Take \(z_0=1+\varepsilon u_0\) which satisfies
By Theorem 2.7, we have that \(z_0 x,z_0 y \in B_E\) and:
Denote
so we have
Notice that using Lemma 3.2, we have
so we obtain
so
Bearing in mind that \(\Vert f\Vert _\mathcal {B}\le \Vert f\Vert _\mathcal {I}\) (see (1.6)), we have
so
and we conclude \(C \le 6 \Vert x\Vert \Vert f\Vert _\mathcal {I}\rho _E(z_0 x,z_0y)\).
Finally, we apply inequality (3.2), and since \(|z_0| \le \frac{1+\Vert x\Vert }{2\Vert x\Vert }\), we obtain
and we are done. \(\square \)
Theorem 3.4
Let \(f \in \mathcal {B}(B_E)\) and \(x,y \in B_E\). Then
Proof
We call
and suppose without loss of generality that \(\Vert x\Vert \ge \Vert y\Vert \). We have that
Since \(\Vert x\Vert ^2-\Vert y\Vert ^2=(\Vert x\Vert +\Vert y\Vert )(\Vert x\Vert -\Vert y\Vert ) \le 2 (\Vert x\Vert -\Vert y\Vert )\) and bearing in mind that \(\rho _E(\Vert x\Vert ,\Vert y\Vert ) \le \rho _E(x,y)\), we obtain
By Lemma 3.3, we know that
so from (3.5), we conclude
and we are done. \(\square \)
Hence, we obtain the result which proves the Lipschitz continuity of the mapping \(x \mapsto (1-\Vert x\Vert ^2)|Rf(x)|\) for \(x \in B_E\).
Corollary 3.5
Let E be a complex Hilbert space. The function \(x \mapsto (1-\Vert x\Vert ^2) |\mathcal {R}f(x)|\) for \(x \in B_E\) is Lipschitz with respect to the pseudohyperbolic distance and the following inequality holds:
Proof
Applying Theorem 3.4, it is clear that
and we are done. \(\square \)
3.2 Results on interpolating sequences for the Bloch space
Recall that a sequence \((x_n) \subset B_E {\setminus } \{0\}\) is said to be interpolating for the Bloch space \(\mathcal {B}(B_E)\) if, for any bounded sequence \((a_n)\) of complex numbers, there exists \(f \in \mathcal {B}(B_E)\) such that \((1-\Vert x_n\Vert ^2) \mathcal {R}f(x_n)=a_n\). Attele studied in [2] this kind of interpolation for the classical Bloch space \(\mathcal {B}\) and the finite- and infinite-dimensional setting was studied in [4]. We provide a new approach to prove that a necessary condition for a sequence \((x_n) \subset B_E\) to be interpolating for \(\mathcal {B}(B_E)\) is to be separated for the pseudohyperbolic distance \(\rho _E\).
Proposition 3.6
Let E be a complex Hilbert space. If \((x_n) \subset B_E {\setminus } \{0\}\) is interpolating for \(\mathcal {B}(B_E)\), then there exists \(C>0\), such that \(\rho (x_k,x_j) \ge C\) for any \(k \ne j\), \(k,j \in \mathbb {N}\).
Proof
Since \((x_n) \subset B_E {\setminus } \{0\}\) is interpolating, there exists a sequence \((f_n) \subset \mathcal {B}(B_E)\), such that
The operator \(T: \mathcal {B}(B_E) \rightarrow \ell _{\infty }\) given by \(T(f)=((1-\Vert x_n\Vert ^2) \mathcal {R}f(x_n))\) is surjective, so by the Open Mapping Theorem, there exists \(M >0\), such that \(\Vert f\Vert _R \le M \sup _{j \in \mathbb {N}} (1-\Vert x_j\Vert ^2) |\mathcal {R}f(x_j)|\), so \(\Vert f_n\Vert _R \le M\) for any \(n \in \mathbb {N}\). Applying Theorem 3.4, we have
Hence, \(1-0 \le 14 A_0 M \rho _E(x_k,x_j)\), and we conclude that
so we are done. \(\square \)
Attele (see [2]) also proved that any interpolating sequence \((z_n) \subset \textbf{D}\) for \(\mathcal {B}\) can be slightly perturbed and the sequence remains interpolating. By means of Theorem 3.4, we adapt his proof and generalize the result to the case when we deal with any complex Hilbert space E.
Theorem 3.7
If \((x_n) \subset B_E {\setminus } \{ 0 \}\) is an interpolating sequence for \(\mathcal {B}(B_E)\), then there exists \(\delta >0\), such that if \((y_n) \subset B_E \) satisfies that \(\sup _{n \in \mathbb {N}} \rho _E(x_n,y_n)<\delta \), then \((y_n)\) is also an interpolating sequence for \(\mathcal {B}(B_E)\).
Proof
Since \((x_n)\) is interpolating, the operator \(T: \mathcal {B}(B_E) \rightarrow \ell _\infty \) given by \(T(f)=((1-\Vert x_n\Vert ^2) \mathcal {R}f(x_n))\) is surjective. Hence, its adjoint \(T^*: \ell _{\infty }^* \rightarrow (\mathcal {B}(B_E))^*\) is injective and it has closed range. In particular, \(T^*\) is left-invertible. The set of left-invertible elements is open in the Banach algebra of linear operators from \(\ell _{\infty }^*\) to \((\mathcal {B}(B_E))^*\). Therefore, there exists \(\delta \), such that if \(\Vert T^*-R\Vert < 14 A_0 \delta \), then R is left-invertible. If we consider \(S(f)=((1-\Vert y_n\Vert ^2) \mathcal {R}f(y_n))\), then by Theorem 3.4
so \(\Vert T-S\Vert < 14 A_0 \delta \), and hence, \(\Vert T^*-S^*\Vert =\Vert T-S\Vert < 14 A_0 \delta \). We conclude that \(S^*\) is left-invertible, and hence, S is surjective, as we wanted. \(\square \)
Data availability
Not applicable.
References
Aron, R.M., Galindo, P., Lindström, M.: Connected components in the space of composition operators of \(H^\infty \) functions of many variables. Integr. Equ. Oper. Theory 45, 1–14 (2003)
Attele, K.R.M.: Interpolating sequences for the derivatives of the Bloch functions. Glasgow Math. J. 34, 35–41 (1992)
Blasco, O., Galindo, P., Lindström, M., Miralles, A.: Composition operators on the Bloch space of the unit ball of a Hilbert space. Banach J. Math. Anal. 11(2), 311–334 (2017)
Blasco, O., Galindo, P., Lindström, M., Miralles, A.: Interpolating sequences for weighted spaces of analytic functions on the unit ball of a Hilbert space. Rev. Mat. Complut. 32(1), 115–139 (2019)
Blasco, O., Galindo, P., Miralles, A.: Bloch functions on the unit ball of an infinite dimensional Hilbert space. J. Funct. Anal. 267, 1188–1204 (2014)
Chen, S.L., Kalaj, D.: Lipschitz continuity of Bloch type mappings with respect to Bergman metric. Ann. Acad. Sci. Fenn. Math. 43, 239–246 (2018)
Chen, S.L., Hamada, H., Zhu, J.-F.: Composition operators on Bloch and Hardy type spaces. Math. Z. 301, 3939–3957 (2022)
Ghatage, P., Yan, J., Zheng, D.: Composition operators with closed range on the Bloch space. Proc. Am. Math. Soc. 129(7), 2039–2044 (2000)
Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker Inc, New York (1984)
Hamada, H.: Distortion theorems, Lipschitz continuity and their applications for Bloch type mappings on bounded symmetric domains in \({\mathbb{C} }^{n}\). Ann. Acad. Sci. Fenn. Math. 44, 1003–1014 (2019)
Miralles, A.: Bounded below composition operators on the space of Bloch functions on the unit ball of a Hilbert space. Banach J. Math. Anal. 17, 73 (2023)
Mujica, J.: Complex Analysis in Banach Spaces, Math. Studies, vol. 120. North-Holland, Amsterdam (1986)
Timoney, R.M.: Bloch functions in several complex variables I. Bull. Lond. Math. Soc. 12, 241–267 (1980)
Xiong, C.: On the Lipschitz continuity of the dilation of Bloch functions. Period. Math. Hung. 47(1–2), 233–238 (2003)
Acknowledgements
This work was supported by PID2019-106529GB-I00 (MICINN, Spain).
Funding
Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author has not competing interests to declare that are relevant to the content of the article.
Additional information
Communicated by Pedro Tradacete.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Miralles, A. Lipschitz continuity of the dilation of Bloch functions on the unit ball of a Hilbert space and applications. Ann. Funct. Anal. 15, 17 (2024). https://doi.org/10.1007/s43034-024-00317-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s43034-024-00317-0