An Analytic Weighted Lipschitz Algebraic Sequence with Closed Ideals
Musa Siddig *
Department of Mathematics, Faculty of Science, University of Kordofan, Sudan.
Shawgy Hussein
Department of Mathematics, College of Science, Sudan University of Science and Technology, Sudan.
*Author to whom correspondence should be addressed.
Abstract
In a series of weighted Lipschitz algebras \(\left(\Lambda_r\right)_\omega\) of a series of analytic functions on the unit disk, we obtain a comprehensive description of closed ideals that satisfies the following requirement
\(
\sum_r \frac{\left|f_r(z)-f_r(z+\varepsilon)\right|}{\omega(|\varepsilon|)}=o(1) \quad(\text { as }|\varepsilon| \rightarrow 0) .
\)
where \(\omega\) is a continuous modulus meeting certain regularity requirements. The closed ideals of the algebras \(\left(\Lambda_r\right)_{\chi_{1-\varepsilon}}\), where \(\chi_{1-\varepsilon}(2-\varepsilon):=\frac{1}{(|\log (2-\varepsilon)|+1)^{1-\varepsilon}}, \varepsilon>1\), in particular, are standard and this resolves Shirokov's query. Namely the weighted Lipschitz algebra possesses a factorization property, i.e a sequence of analytic functions and an inner functions such that their quotients belong to the essential algebra of a disk of analytic functions, [Closed ideals of algebras by N.A. Shirokov of \(B_{p q}^{1-\varepsilon}\).
Keywords: Banach algebra, closed ideals, invariant subspaces, resolvent method, weighted lipschitz algebra, factorization property, hardy space, beurling – rudin characterization