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Subnormality and composition operators on the Bergman space. (English) Zbl 1041.47010

C. Cowen and T. Kriete [J. Funct. Anal. 81, 298–319 (1988; Zbl 0669.47012)] characterized the analytic maps \(\varphi: \mathbb{D}\to \mathbb{D}\) (\(\mathbb{D}=\) the unit disk) for which the composition operator \(C_\varphi \) defined by \(C_\varphi f =f \circ \varphi \) on the Hardy space \(H^2(\mathbb{D})\) has a subnormal adjoint. Specifically, they showed that under a regularity condition near the Denjoy-Wolff point \(d\), \(| d| =1\), \(C_\varphi^*\) is subnormal on \(H^2\) if and only if \[ \varphi (z) = \frac{(r+s)z+(1-s)d}{r(1-s)\overline{d}z+(1+sr)} \] where \(0< s = \varphi ' (d) < 1 \) and \(0 \leq r \leq 1\). C. Cowen [Integral Equations Oper. Theory 15, 167–171 (1992; Zbl 0773.47009)] proved that if for \(\varphi \), \(C_\varphi^*\) is subnormal on \(H^2\), then \(\varphi \) also gives rise to a subnormal operator \(C_\varphi^*\) on \(A^2(\mathbb{D})\).
In the paper under review, the author characterizes those analytic selfmaps \(\varphi\) of \(\mathbb{D}\), for which \(C_\varphi^*\) is subnormal on \(A^2(\mathbb{D})\). The conditions are as those in the Hardy space case but with large range for the parameter \(r\); in fact, \(r\) is allowed in \([ - 1/7, 1]\).

MSC:

47B33 Linear composition operators
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
47B20 Subnormal operators, hyponormal operators, etc.
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