Duality for vector-valued Bergman–Orlicz spaces and little Hankel operators between vector-valued Bergman–Orlicz spaces on the unit ball

In this paper, we consider vector-valued Bergman–Orlicz spaces which are generalization of classical vector-valued Bergman spaces. We characterize the dual space of vector-valued Bergman–Orlicz space, and study the boundedness of the little Hankel operators, \(h_b\) $h_b$ , with operator-valued symbols b, between different weighted vector-valued Bergman–Orlicz spaces on the unit ball \(\mathbb{B}_n\) ${\mathbb{B}}_n$ .More precisely, given two complex Banach spaces X, Y, we characterize those operator-valued symbols \(b \colon \mathbb{B}_n\rightarrow \mathcal{L} (\overline{X},Y) \) $b:{\mathbb{B}}_n\to \mathcal{L}(\overline{X},Y)$ for which the little Hankel operator \(h_{b}: A^{\Phi_{1}}_{\alpha}(\mathbb{B}_{n},X) \longrightarrow A^{\Phi_{2}}_{\alpha}(\mathbb{B}_{n},Y)\) $h_b:A_{\alpha }^{{\mathrm{\Phi }}_1}({\mathbb{B}}_n,X)⟶A_{\alpha }^{{\mathrm{\Phi }}_2}({\mathbb{B}}_n,Y)$ , extends into a bounded operator, where \(\Phi_{1}\) ${\mathrm{\Phi }}_1$ and \(\Phi_2\) ${\mathrm{\Phi }}_2$ are either convex or concave growth functions.