12 . 5 . BLA3CHKE PRODUCTS AND IDEALS I^ CT-

Let A be the space of functions analytic in the open unit disc D and continuous in OÏOJ D ; and let C^*t^^ • Г ^^A, 1^=0, i,...J• Although the sets of uniqueness for Ca have been described [l] ,[2] , [з],[4] , and the closed ideal structure of C^ is known [з], there are still some open questions concerning the relationship of KLaschke products with closed ideals in Сд • I pose two problems. Let I ,I^C^ , denote a closed ideal and let В denote a Blaschke product which divides some non-zero Ca ftinction.

( 1 ) P 0 r w h i с .h В is it^true that

( 2 ) Ifß is the g.c.d. (greatest common sor) of the Blaschke factors of the

n 0 n .^ z e r^o functions iri ,when is СУВ)! — l^^^A ' ß^^IJ a closed ideal inC^?

Note that the corresponding problems for singralar inner tions are easier and are solved in section 4 of [5]«

To discuss the problems for Blaschke products we need some notation. Let

and let Z'XL)-[['^(l)andi Z(I) = i^"'(J)irv=o • If I(Z(I))denotes the closed idesO^of all f ♦ f ^Сд , with f^"'Чх) =0 for t^Z CD» tv-OJv» then the closed ideal structure theorem says I* S'I (Z (I)) where S is the g.c.d. of the singular inner tors of the non-zero fimctions in X

DEFINITION . A sequence [XjJ cQ has finite degree of contact atE»E c9D , if there exist к ,k>0 , ande , e>0 , such that l-IZjI^epC^j/IZjl, E)*" for all j • (Here p denotes the Euclidean metric.)

übe following unpublished theorem of B.A.Taylor and the author provides solutions to problems (l) and (2) in a special , case « ^ oe oo

THEOREM , (a) Assume ä(I)~Z(I). In order that о1^Сд it is necessary and sufficient that the zeros of В have finite degree of contact at Z**(I). If ВХ^^Сд, then multiplication by В is continuous on I ,61 is closed, and the inverse tion is continuous« 00

( b ) Assume Z^'QrflO^ZCD- Let 6 be the g.c.d. of the

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