Litte Hankel Operators Between Vector-Valued Bergman Spaces on the Unit Ball
Tóm tắt
In this paper, we study the boundedness and the compactness of the little Hankel operators
$$h_b$$
with operator-valued symbols b between different weighted vector-valued Bergman spaces on the open unit ball
$$\mathbb {B}_n$$
in
$$\mathbb {C}^n.$$
More precisely, given two complex Banach spaces X, Y, and
$$0 < p,q \le 1,$$
we characterize those operator-valued symbols
$$b: \mathbb {B}_{n}\rightarrow \mathcal {L}(\overline{X},Y)$$
for which the little Hankel operator
$$h_{b}: A^p_{\alpha }(\mathbb {B}_{n},X) \longrightarrow A^q_{\alpha }(\mathbb {B}_{n},Y),$$
is a bounded operator. Also, given two reflexive complex Banach spaces X, Y and
$$1< p \le q < \infty ,$$
we characterize those operator-valued symbols
$$b: \mathbb {B}_{n}\rightarrow \mathcal {L}(\overline{X},Y)$$
for which the little Hankel operator
$$h_{b}: A^p_{\alpha }(\mathbb {B}_{n},X) \longrightarrow A^q_{\alpha }(\mathbb {B}_{n},Y),$$
is a compact operator.
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