Abstract
AbstractLet $$B_E$$
B
E
be the open unit ball of a complex finite- or infinite-dimensional Hilbert space. If f belongs to the space $$\mathcal {B}(B_E)$$
B
(
B
E
)
of Bloch functions on $$B_E$$
B
E
, we prove that the dilation map given by $$x \mapsto (1-\Vert x\Vert ^2) \mathcal {R}f(x)$$
x
↦
(
1
-
‖
x
‖
2
)
R
f
(
x
)
for $$x \in B_E$$
x
∈
B
E
, where $$\mathcal {R}f$$
R
f
denotes the radial derivative of f, is Lipschitz continuous with respect to the pseudohyperbolic distance $$\rho _E$$
ρ
E
in $$B_E$$
B
E
, which extends to the finite- and infinite-dimensional setting the result given for the classical Bloch space $$\mathcal {B}$$
B
. To provide this result, we will need to prove that $$\rho _E(zx,zy) \le |z| \rho _E(x,y)$$
ρ
E
(
z
x
,
z
y
)
≤
|
z
|
ρ
E
(
x
,
y
)
for $$x,y \in B_E$$
x
,
y
∈
B
E
under some conditions on $$z \in \mathbb {C}$$
z
∈
C
. Lipschitz continuity of $$x \mapsto (1-\Vert x\Vert ^2) \mathcal {R}f(x)$$
x
↦
(
1
-
‖
x
‖
2
)
R
f
(
x
)
will yield some applications on interpolating sequences for $$\mathcal {B}(B_E)$$
B
(
B
E
)
which also extends classical results from $$\mathcal {B}$$
B
to $$\mathcal {B}(B_E)$$
B
(
B
E
)
. Indeed, we show that it is necessary for a sequence in $$B_E$$
B
E
to be separated to be interpolating for $$\mathcal {B}(B_E)$$
B
(
B
E
)
and we also prove that any interpolating sequence for $$\mathcal {B}(B_E)$$
B
(
B
E
)
can be slightly perturbed and it remains interpolating.
Publisher
Springer Science and Business Media LLC