Tu, Zhenhan; Xiong, Liangpeng Weighted space and Bloch-type space on the unit ball of an infinite dimensional complex Banach space. (English) Zbl 07120122 Bull. Iran. Math. Soc. 45, No. 5, 1389-1406 (2019). Summary: Let \(\mathbf{B}_{\mathbb{X}}\) be the open unit ball of a complex Banach space \(\mathbb{X}\), which may beinfinite dimensional. The weighted composition operator and weighted space defined on \(\mathbf{B}_{\mathbb{X}}\) are introduced. We obtain the boundedness and compactness of the weighted composition operator from the Bloch-type spaces to the weighted spaces, and some properties with the Bloch-type spaces are given. Our main results generalize theprevious works on the Euclidean unit ball \(\mathbb{B}^n\) to the case of \(\mathbf{B}_{\mathbb{X}}\). Cited in 2 Documents MSC: 47B38 Linear operators on function spaces (general) 32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) 46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces Keywords:boundedness; complex Banach space; compactness; weighted composition operator; weighted Bloch-type space PDFBibTeX XMLCite \textit{Z. Tu} and \textit{L. Xiong}, Bull. Iran. Math. Soc. 45, No. 5, 1389--1406 (2019; Zbl 07120122) Full Text: DOI References: [1] Anderson, J.M., Clunie, J.G., Pommerenke, Ch.: On Bloch functions and normal functions. J. Reine Angew. Math. 270, 12-37 (1974) · Zbl 0292.30030 [2] Allen, R.F., Colonna, F.: Weighted composition operators from \[H^{\infty }H\]∞ to the Bloch space of a bounded homogeneous domain. Integr. Equ. Oper. 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