An Analytic Weighted Lipschitz Algebraic Sequence with Closed Ideals
Journal of Advances in Mathematics and Computer Science · pp. 100–121 · Published 6 May 2023
10.9734/jamcs/2023/v38i71777Abstract
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 \(B_{p q}^{1-\varepsilon}\).
References (6)
- 1 Banach Spaces of Analytic Functions. [DOI]
- 2 INVARIANT SUBSPACES OF THE SHIFT OPERATOR IN WEIGHTED HILBERT SPACE [DOI]
- 3 Analytic Functions Smooth up to the Boundary [DOI]
- 4 Closed ideals in analytic weighted Lipschitz algebras [DOI]
- 5 Cyclic vectors for invariant subspaces in some classes of analytic functions [DOI]
- 6 CLOSED IDEALS IN SOME ALGEBRAS OF ANALYTIC FUNCTIONS
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