3,115 research outputs found

    The Commutant of Multiplication by z on the Closure of Rational Functions in Lt(μ)L^t(\mu)

    Full text link
    For a compact set KC,K\subset \mathbb C, a finite positive Borel measure μ\mu on K,K, and 1 \le t < \i, let Rat(K)\text{Rat}(K) be the set of rational functions with poles off KK and let Rt(K,μ)R^t(K, \mu) be the closure of Rat(K)\text{Rat}(K) in Lt(μ).L^t(\mu). For a bounded Borel subset DC,\mathcal D\subset \mathbb C, let \area_{\mathcal D} denote the area (Lebesgue) measure restricted to D\mathcal D and let H^\i (\mathcal D) be the weak-star closed sub-algebra of L^\i(\area_{\mathcal D}) spanned by f,f, bounded and analytic on CEf\mathbb C\setminus E_f for some compact subset EfCD.E_f \subset \mathbb C\setminus \mathcal D. We show that if Rt(K,μ)R^t(K, \mu) contains no non-trivial direct LtL^t summands, then there exists a Borel subset RK\mathcal R \subset K whose closure contains the support of μ\mu and there exists an isometric isomorphism and a weak-star homeomorphism ρ\rho from Rt(K,μ)L(μ)R^t(K, \mu) \cap L^\infty(\mu) onto H(R)H^\infty(\mathcal R) such that ρ(r)=r\rho(r) = r for all rRat(K).r\in\text{Rat}(K). Consequently, we obtain some structural decomposition theorems for \rtkmu.Comment: arXiv admin note: text overlap with arXiv:2212.1081

    Invertibility in Weak-Star Closed Algebras of Analytic Functions

    Full text link
    For KCK\subset \mathbb C a compact subset and μ\mu a positive finite Bore1 measure supported on K,K, let R(K,μ)R^\infty (K,\mu) be the weak-star closure in L(μ)L^\infty (\mu) of rational functions with poles off K.K. We show that if R(K,μ)R^\infty (K,\mu) has no non-trivial LL^\infty summands and fR(K,μ),f\in R^\infty (K,\mu), then ff is invertible in R(K,μ)R^\infty (K,\mu) if and only if Chaumat's map for KK and μ\mu applied to ff is bounded away from zero on the envelope with respect to KK and μ.\mu. The result proves the conjecture \diamond posed by J. Dudziak in 1984.Comment: arXiv admin note: text overlap with arXiv:2212.1081