Validity of Closed Ideals in Algebras of Series of Square Analytic Functions

Musa Siddig,Amani Elseid,Shawgy Hussein

doi:10.22158/asir.v5n1p20

Musa Siddig, Amani Elseid + Show 1 more

Open Access

https://doi.org/10.22158/asir.v5n1p20

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Abstract

We show the validity of a complete description of closed ideals of the algebra which is a commutative Banach algebra , that endowed with a pointwise operations act on Dirichlet space of algebra of series of analytic functions on the unit disk satisfying the Lipscitz condition of order of square sequence obtained by (Brahim Bouya, 2008), we introduce and deal with approximation square functions which is an outer functions to produce and show results in .

Highlights

The Dirichlet space D consists of the sequence of square complex-valued analytic functions fj2 on the unit disk D with finite Dirichlet integral∑ D(fj2 ) : = ∫ ∑ |(fj2)′(z)|2 dA(z) < +∞, jDj where dA(z) = 1 (1 − ε)d(1 − ε)dt2 π denotes the normalized area measure onD
Banach algebra Aαj2, that endowed with a pointwise operations act on Dirichlet space of algebra of series of analytic functions on the unit disk D satisfying the Lipscitz condition of order of square sequence αj2 obtained by (Brahim Bouya, 2008), we introduce and deal with approximation square functions which is an outer functions to produce and show results in Aαj2
Brahim Bouya (2008) described the structure of the closed ideals of the Banach algebras Aαj2. More precisely he proved that these ideals are standard in the sense of the Beurling-Rudin characterization of the closed ideals in the disc algebra (Hoffman, 1988), we show the general validation following (Brahim Bouya, 2008): Theorem (1.1): If I is closed ideal of Aαj2

Summary

Introduction

(1972) has described the closed ideals of the algebra H12 of sequence of square analytic functions fj2 such that (fj2)′ ∈ H2, where H2 is the Hardy space. Brahim Bouya (2008) described the structure of the closed ideals of the Banach algebras Aαj2. More precisely he proved that these ideals are standard in the sense of the Beurling-Rudin characterization of the closed ideals in the disc algebra (Hoffman, 1988), we show the general validation following (Brahim Bouya, 2008): Theorem (1.1): If I is closed ideal of Aαj2, Theorem (1.2): Let fj2 be a function in Aαj2\ {0} and let ε ≥ 0.

Results

Conclusion