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Composition operators on weighted Bergman–Orlicz spaces. (English) Zbl 1119.47024

The authors consider composition operators acting on the weighted Bergman–Orlicz space \(A^\Phi_\alpha\) of analytic functions in the unit disk such that \(\int_D \Phi(\log| f(z)| ) d\nu_\alpha(z)<\infty\) where, as usual, \(d\nu_\alpha(z)=(\alpha+1)(1-| z| ^2)^\alpha dA(z)\) and \(\Phi\) is a non-constant, non-decreasing convex function defined on \((-\infty, \infty)\) and \(\lim_{t\to\infty} \Phi(t)/t=\infty\). They prove that \(C_\varphi\) is bounded on \(A^\Phi_\alpha\) if and only if the pull-back measure \(\mu_{\phi,\alpha}= \nu_\alpha\circ \varphi^{-1}\) is an \(\alpha\)-Carleson measure. Also, the compactness is characterized by the pull-back measure being a vanishing \(\alpha\)-Carleson measure.
Since the same results hold true from the work of B.D.MacCluer and J.H.Shapiro [Can.J.Math.38, 878–906 (1986; Zbl 0608.30050)] in the \(A^p_\alpha\) case, their result implies that any self-map defines a bounded operator on \(A^\Phi_\alpha\) and that it is compact if and only if it is compact on \(A^2_\alpha\).
The same results are also obtained in the more general setting of generalized Bergman spaces which have also been considered by S.Stević [Complex Variables, Theory Appl.49, No.2, 109–124 (2004; Zbl 1053.47020)].

MSC:

47B33 Linear composition operators
46E10 Topological linear spaces of continuous, differentiable or analytic functions
30D55 \(H^p\)-classes (MSC2000)
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References:

[1] MacCluer, Canad. J. Math. 38 pp 878– (1986) · Zbl 0608.30050 · doi:10.4153/CJM-1986-043-4
[2] DOI: 10.1007/BF01198631 · Zbl 0657.30026 · doi:10.1007/BF01198631
[3] Cima, Pacific J. Math. 179 pp 59– (1997)
[4] Axler, Surveys of some recent results in operator theory, Vol. 1 171 pp 1– (1988)
[5] DOI: 10.2307/2048293 · Zbl 0704.47018 · doi:10.2307/2048293
[6] Stevic, Complex Var. Theory. Appl. 49 pp 109– (2004)
[7] DOI: 10.2307/1971314 · Zbl 0642.47027 · doi:10.2307/1971314
[8] DOI: 10.1512/iumj.1973.23.23041 · Zbl 0276.47037 · doi:10.1512/iumj.1973.23.23041
[9] Shapiro, Composition operators and classical function theory (1993) · Zbl 0791.30033 · doi:10.1007/978-1-4612-0887-7
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