We characterize boundedness and compactness of the classical Volterra operator Tg : H ∞ vα → H ∞ induced by a univalent function g for standard weights vα with 0 ≤ α < 1, partly answering an open problem posed by A. Anderson, M. Jovovic and W. Smith. We also study boundedness, compactness and weak compactness of the generalized Volterra operator T ϕ g mapping between Banach spaces of analytic functions on the unit disc satisfying certain general conditions.2010 Mathematics Subject Classification. Primary 47B38, Secondary 46B50.
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Very recently, Božin and Karapetrović [4] solved a conjecture by proving that the norm of the Hilbert matrix operator H on the Bergman space A p is equal to π sin( 2π p ) for 2 < p < 4. In this article we present a partly new and simplified proof of this result. Moreover, we calculate the exact value of the norm of H defined on the Korenblum spaces H ∞ α for 0 < α ≤ 2/3 and an upper bound for the norm on the scale 2/3 < α < 1.
In this article, the open problem of finding the exact value of the norm of the Hilbert matrix operator on weighted Bergman spaces A p α is adressed. The norm was conjectured to be π sin (2+α)π p by Karapetrović. We obtain a complete solution to the conjecture for α > 0 andand a partial solution for 2 + 2α < p < 2 + α + α 2 + 7 2 α + 3. Moreover, we also show that the conjecture is valid for small values of α when 2 + 2α < p ≤ 3 + 2α. Finally, the case α = 1 is considered.
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