Geometric constructions of thin Blaschke products and reducing subspace problem. Issue 4 (19th June 2014)

• Record Type: Journal Article
• Authors: Guo, Kunyu; Huang, Hansong
• Is Part Of: Proceedings of the London Mathematical Society. Volume 109:Issue 4(2014:Oct.)
• Page Start: 1050
• Page End: 1091
• Publication Date: 2014-06-19
• Subjects: Mathematics -- Periodicals; Mathematics; Periodicals; 510
• DOI: 10.1112/plms/pdu027
• Languages: English
• ISSNs: 0024-6115

Abstract

Abstract : In this paper, we mainly study geometric constructions of thin Blaschke productsB and reducing subspace problem of multiplication operators induced by such symbolsB on the Bergman space. Considering such multiplication operatorsM B, we present a representation of those operators commuting with bothM B andM B * . It is shown that for 'most' thin Blaschke productsB, M B is irreducible, that is, M B has no nontrivial reducing subspace; and such a thin Blaschke productB is constructed. As an application of the methods, it is proved that for 'most' finite Blaschke productsϕ, M ϕ has exactly two minimal reducing subspaces. Furthermore, under a mild condition, we get a geometric characterization for whenM B defined by a thin Blaschke productB has a nontrivial reducing subspace.