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Litte Hankel Operators Between Vector-Valued Bergman Spaces on the Unit Ball

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Abstract

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 XY,  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 XY 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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Correspondence to Edgar L. Tchoundja.

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The second author would like to acknowledge the support of the GRAID program of IMU/CDC. He would also like to thank the International Centre for Theoretical Physic (ICTP), Trieste (Italy) for partially supporting our visit to the centre where we have progressed in this work. B. D. Wick’s research partially supported in part by NSF Grant DMS-1800057 as well as ARC DP190100970.

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Békollé, D., Defo, H.O., Tchoundja, E.L. et al. Litte Hankel Operators Between Vector-Valued Bergman Spaces on the Unit Ball. Integr. Equ. Oper. Theory 93, 28 (2021). https://doi.org/10.1007/s00020-021-02640-w

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