Composition operators on the normal weight Zygmund spaces in high dimensions. (English) Zbl 1442.47020
Summary: Let \(\mu\) be a normal function on \([0,1)\). Suppose that \(\varphi\) is a holomorphic self-map on the unit ball \(B\) of \(\mathbb{C}^n\). In this paper, the authors give a new kind of equivalent norm of the normal weight Zygmund space \(Z_\mu(B)\). On the basis of their recent work [Complex Var. Elliptic Equ. 64, No. 11, 1932–1953 (2019; Zbl 1425.32006)], the authors further study boundedness criteria of composition operators \(C_\varphi\) on \(Z_\mu(B)\) and the essential norm of \(C_\varphi\) for all normal weights \(\mu\). When \(\int_0^1 \int_0^r (dt/ \mu(t))\,dr < \infty\) (or \(\int_0^1 (dr/ \mu(r)) < \infty)\), the authors further give a brief compactness criteria of \(C_\varphi\) on \(Z_\mu(B)\) (or \(\beta_\mu(B))\).
MSC:
| 47B33 | Linear composition operators |
| 46E15 | Banach spaces of continuous, differentiable or analytic functions |
| 32A37 | Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) |
Keywords:
normal weighted Zygmund space; composition operator; boundedness; compactness; essential normCitations:
Zbl 1425.32006References:
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