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Composition operators with maximal norm on weighted Bergman spaces. (English) Zbl 1110.47016

It is well-known that every analytic self-map of the unit disk induces a bounded composition operator on the Hardy space \(H^2\). Moreover, the well-known Littlewood’s subordination theorem implies \[ \| C_\varphi\| ^2_{H^2\to H^2}\leq {1+| \varphi(0)| \over 1-| \varphi(0)| }. \] It was shown in E.A.Nordgren’s early paper in this field [Can.J.Math.20, 442–449 (1968; Zbl 0161.34703)] that for inner functions the norm of \(C_\varphi\) is maximal, i.e., equals the upper bound given above. More recently, J.H.Shapiro [Monatsh.Math.130, No.1, 57–70 (2000; Zbl 0951.47026)] showed that whenever \(\varphi(0) \neq 0\), the norm is maximal only when \(\varphi\) is an inner function.
In the paper under review, the authors prove an analogue of this maximality result in the context of weighted Bergman spaces \(A^2_\alpha\), \(\alpha>-1\). Recall that every composition operator is also bounded on each \(A^2_\alpha\) and the upper bound for its norm is similar: \[ \| C_\varphi\| ^2_{A^2_\alpha\to A^2_\alpha} \leq \bigg({1+| \varphi(0)| \over 1-| \varphi(0)| }\bigg)^{2+\alpha}. \] The authors prove that, whenever \(\varphi(0) \neq 0\), the norm of the composition operator \(C_\varphi\) acting on \(A^2_\alpha\) equals the above upper bound if and only if \(\varphi\) is a disk automorphism. They show that this is also equivalent to the maximality of the essential norm of the operator.

MSC:

47B33 Linear composition operators
30H05 Spaces of bounded analytic functions of one complex variable
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