Reducing subspaces for analytic multipliers of the Bergman space (Q452370)

The paper under review aims at the classification of the reducing subspaces of multiplication operators \(M_\phi\) with a rational, inner symbol \(\phi,\) acting on the Bergman space of the unit disc \(L^2_a({\mathbb D})\). The reducing subspaces of \(M_\phi\) are the ranges of the projections in the commutant \(\{M_\phi\}^\prime\).NEWLINENEWLINEThe same classification problem for analytic multipliers on the Hardy space was settled in the 1970s by \textit{C. C. Cowen} [Trans. Am. Math. Soc. 239, 1--31 (1978; Zbl 0391.47014)] and \textit{J. E. Thomson} [Indiana Univ. Math. J. 25, 793--800 (1976; Zbl 0334.47023); Am. J. Math. 99, 522--529 (1977; Zbl 0372.47018)]. They obtained a description of the reducing subspaces of \(M_\phi\) in terms of the Riemann surface of \(\phi^{-1} \circ \phi\).NEWLINENEWLINERecently, the first author, \textit{S.-H. Sun} and \textit{D.-C. Zheng} [Adv. Math. 226, No. 1, 541--583 (2011; Zbl 1216.47053)] showed in the Bergman space context that, for a finite Blaschke product \(\phi,\) the double commutant \(\{M_\phi,M_\phi^\ast\}^\prime\) is finite dimensional and its dimension equals the number of connected components of the Riemann surface \(\phi^{-1} \circ \phi\). They also conjectured that the double commutant \(\{M_\phi,M_\phi^\ast\}^\prime\) is an abelian algebra. The main result in the paper under review gives an affirmative answer to this question. As a corollary, \(M_\phi\) has exactly \(q\) minimal reducing subspaces, where \(q\) equals the number of connected components of \(\phi^{-1} \circ \phi\). Furthermore, the minimal reducing subspaces of \(M_\phi\) are pairwise orthogonal. The authors also provide an arithmetic description of the minimal reducing subspaces of \(M_\phi\) and a taxonomy of the reducing subspaces when \(\phi\) has eight zeros.