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WEAKLY DENSE IDEALS IN PRIVALOV SPACES OF HOLOMORPHIC FUNCTIONS
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 Title & Authors
WEAKLY DENSE IDEALS IN PRIVALOV SPACES OF HOLOMORPHIC FUNCTIONS
Mestrovic, Romeo; Pavicevic, Zarko;
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 Abstract
In this paper we study the structure of closed weakly dense ideals in Privalov spaces (1 < p < ) of holomorphic functions on the disk : |z| < 1. The space with the topology given by Stoll's metric [21] becomes an F-algebra. N. Mochizuki [16] proved that a closed ideal in is a principal ideal generated by an inner function. Consequently, a closed subspace E of is invariant under multiplication by z if and only if it has the form for some inner function I. We prove that if is a closed ideal in that is dense in the weak topology of , then is generated by a singular inner function. On the other hand, if is a singular inner function whose associated singular measure has the modulus of continuity , then we prove that the ideal is weakly dense in . Consequently, for such singular inner function , the quotient space is an F-space with trivial dual, and hence does not have the separation property.
 Keywords
Privalov space ;F-algebra;weakly dense ideal;singular inner function;topological dual;
 Language
English
 Cited by
1.
Topological and Functional Properties of SomeF-Algebras of Holomorphic Functions, Journal of Function Spaces, 2015, 2015, 1  crossref(new windwow)
2.
OnF-AlgebrasMp  (1crossref(new windwow)
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