Hilbert Matrix and Its Norm on Weighted Bergman Spaces
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Hilbert Matrix and Its Norm on Weighted Bergman Spaces Boban Karapetrovi´c1 Received: 2 April 2020 / Accepted: 2 September 2020 © Mathematica Josephina, Inc. 2020
Abstract It is well known that the Hilbert matrix H is bounded on weighted Bergman spaces p . The Aα if and only if 1 < α + 2 < p with the conjectured norm π/ sin (α+2)π p conjecture was confirmed in the case when α = 0 and also in the case when α > 0 and p ≥ 2(α + 2), which reduces the conjecture in the case when α > 0 to the interval α + 2 < p < 2(α + 2). In the remaining case when −1 < α < 0 and p > α + 2 there has been no progress so far in proving the conjecture, moreover, there is no even an explicit upper bound for the norm of the Hilbert matrix H on weighted Bergman spaces p Aα . In this paper we obtain results which are better than known related to the validity of the mentioned conjecture in the case when α > 0 and α +2 < p < 2(α +2). On the other hand, we also provide for the first time an explicit upper bound for the norm of p the Hilbert matrix H on weighted Bergman spaces Aα in the case when −1 < α < 0 and p > α + 2. Keywords Hilbert matrix · Norm · Weighted Bergman spaces Mathematics Subject Classification Primary 47B35 · Secondary 30H20
1 Introduction The Hilbert matrix H and its action on the space 2 consisting of square summable sequences was first studied in [11], where Magnus described the spectrum of the Hilbert matrix. Thereafter Diamantopoulos and Siskakis in [3,4] begin to study the action of the Hilbert matrix on Hardy and Bergman spaces, which can be seen as the beginning of studying of the Hilbert matrix as an operator on spaces of holomorphic functions. They obtained some partial results concerning the questions of boundedness
The author was supported in part by Serbian Ministry of Education, Science and Technological Development, Project #174032.
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Boban Karapetrovi´c [email protected] Faculty of Mathematics, University of Belgrade, Studentski trg 16, Beograd, Serbia
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B. Karapetrovi´c
and exact norm of the Hilbert matrix on Hardy and Bergman spaces, which have been improved in [5] by Dostani´c, Jevti´c and Vukoti´c. We note also that Aleman, MontesRodríguez and Sarafoleanu provide a closed formula for the eigenvalues of the Hilbert matrix in a more general context (see [1]). Following the above results, it was known that Hilbert matrix H is bounded on Bergman space A p if and only if 2 < p < ∞ and H A p →A p =
π sin
2π p
,
when 4 ≤ p < ∞. It was also conjectured that previous equality remains valid in the remaining case when 2 < p < 4. This conjecture was actually proven in [2], where the new method based on the new way to use monotonicity of the integral means was introduced (see also [9]). The starting point for studying the boundedness of the Hilbert matrix H on weighted p Bergman spaces Aα was paper [6] by Galanopoulos, Girela, Peláez and Siskakis, where the corresponding partial results were obtained. A complete characterization of the p boundedness of the Hilbert matrix H on the spaces Aα is
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