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We prove that certain maximal ideals in Beurling algebras on the unit disc have approximate identities, and show the existence of functions with certain properties in these maximal ideals. We then use these results to prove that if T is a bounded operator on a Banach space X satisfying $\parallel T^n\parallel =O\left(n^{\beta }\right)$ as n → ∞ for some β ≥ 0, then ${\sum }_{n=1}^{\infty }\parallel {(1-T)}^{n}x\parallel /\parallel {(1-T)}^{n-1}x\parallel$ diverges for every x ∈ X such that ${(1-T)}^{\left[\beta \right]+1}x\ne 0$.

For 0 < γ ≤ 1, let $\Lambda {⁺}_{\gamma }$ be the big Lipschitz algebra of functions analytic on the open unit disc which satisfy a Lipschitz condition of order γ on ̅. For a closed set E on the unit circle and an inner function Q, let $J_{\gamma }(E,Q)$ be the closed ideal in $\Lambda {⁺}_{\gamma }$ consisting of those functions $f\in \Lambda {⁺}_{\gamma }$ for which (i) f = 0 on E, (ii) $|f\left(z\right)-f\left(w\right)|=o\left(\right|z-w{|}^{\gamma })$ as d(z,E),d(w,E) → 0, (iii) $f/Q\in \Lambda {⁺}_{\gamma }$. Also, for a closed ideal I in $\Lambda {⁺}_{\gamma }$, let $E_I$ = z ∈ : f(z) = 0 for every f ∈ I and let $Q_I$ be the greatest common divisor of the inner parts of non-zero functions in I....

Let ${𝒜}_{\beta }$ be the Beurling algebra with weight $(1+|n|{)}^{\beta }$ on the unit circle $𝕋$ and, for a closed set $E\subseteq 𝕋$, let $J_{{𝒜}_{\beta }}\left(E\right)=\{f\in {𝒜}_{\beta }:f=0\text{on}\text{a}\text{neighbourhood}\text{of}E\}$. We prove that, for $\beta \&gt;\frac{1}{2}$, there exists a closed set $E\subseteq 𝕋$ of measure zero such that the quotient algebra ${𝒜}_{\beta }/\overline{J_{{𝒜}_{\beta }}\left(E\right)}$ is not generated by its idempotents, thus contrasting a result of Zouakia. Furthermore, for the Lipschitz algebras ${\lambda }_{\gamma }$ and the algebra $𝒜𝒞$ of absolutely continuous functions on $𝕋$, we characterize the closed sets $E\subseteq 𝕋$ for which the restriction algebras ${\lambda }_{\gamma }\left(E\right)$ and $𝒜𝒞\left(E\right)$ are generated by their idempotents.

We prove that a homogeneous Banach space B on the unit circle T can be embedded as a closed subspace of a dual space Ξ*_B contained in the space of bounded Borel measures on T in such a way that the map B → Ξ*_B defines a bijective correspondence between the class of homogeneous Banach spaces on T and the class of prehomogeneous Banach spaces on T. We apply our results to show that the algebra of all continuous functions on T is the only homogeneous...

For an increasing sequence (ωₙ) of algebra weights on ℝ⁺ we study various properties of the Fréchet algebra A(ω) = ⋂ ₙ L¹(ωₙ) obtained as the intersection of the weighted Banach algebras L¹(ωₙ). We show that every endomorphism of A(ω) is standard, if for all n ∈ ℕ there exists m ∈ ℕ such that ${\omega }_m\left(t\right)/\omega ₙ\left(t\right)\to \infty$ as t → ∞. Moreover, we characterise the continuous derivations on this algebra: Let M(ωₙ) be the corresponding weighted measure algebras and let B(ω) = ⋂ ₙM(ωₙ). If for all n ∈ ℕ there exists m ∈ ℕ such that...