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Interpolating Blaschke Products and Factorization Theorems
Author(s) -
Izuchi Keiji
Publication year - 1994
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/50.3.547
Subject(s) - blaschke product , mathematics , zero (linguistics) , maximal ideal , unit disk , zero set , product (mathematics) , analytic function , factorization , unit (ring theory) , ideal (ethics) , banach algebra , hardy space , bounded function , function (biology) , generalization , space (punctuation) , combinatorics , discrete mathematics , pure mathematics , banach space , mathematical analysis , algorithm , philosophy , linguistics , geometry , mathematics education , epistemology , evolutionary biology , biology
Let M ( H ∞ ) be the maximal ideal space of H ∞ the Banach algebra of bounded analytic functions on the open unit disk. Let G be the set of nontrivial points in M ( H ∞ ). By Hoffman's work, G has deep connections with the zero sets of interpolating Blaschke products. It is proved that for a closed ρ ‐separated subset E of M ( H ∞ ) with E ⊂ G , there exists an interpolating Blaschke product whose zero set contains E . This is a generalization of Lingenberg's theorem. Let f be a continuous function on M ( H ∞ ). Suppose that f is analytic on a nontrivial Gleason part P ( x ), f ( x ) = 0, and f ≠ 0 on P ( x ). It is proved that there is an interpolating Blaschke product b with zeros { z n } n such that b ( x ) = 0 and f ( z n ) = 0 for every n . This fact can be used for factorization theorems in Douglas algebras and in algebras of functions analytic on Gleason parts.