Norm of the Hilbert Matrix Operator on the Weighted Bergman Spaces

Abstract

We find the lower bound for the norm of the Hilbert matrix operator H on the weighted Bergman space Ap,α \begin{equation*} \|H\|_{A^{p,\alpha}\rightarrow A^{p,\alpha}}\geq\frac{\pi}{\sin{\frac{(\alpha+2)\pi}{p}}}, \,\, \textnormal{for} \,\, 1<\alpha+2<p. \end{equation*} We show that if 4 ≤ 2(α + 2) ≤ p, then ∥H∥Ap,α → Ap,α = $\frac{\pi}{\sin{\frac{(\alpha+2)\pi}{p}}}$, while if 2 ≤ α +2 < p < 2(α+2), upper bound for the norm ∥H∥Ap,α → Ap,α, better then known, is obtained.

On the exact value of the norm of the Hilbert matrix operator on weighted Bergman spaces

“…and obtains a better than known upper bound for the norm when 2 ≤ 2 + α < p < 2(2 + α). In [12] Karapetrović conjectures that the norm of H is the same as above also in the case 2 < 2 + α ≤ p < 2(2 + α). In this article the conjecture is confirmed in the positive for…”

“…Proof of Theorem 1.1. Note that the lower bound of the norm of H holds for all α ≥ 0 and all 2 + 2α < p < 2(2 + α) by Theorem 1.1 in [12]. For the upper bound of the norm of H we have by the above argument that if (a) is true, then the conclusion of the theorem holds.…”

“…is the disk with radius 1−t 2−t and center 1 2−t , see [12]. We will also need the Beta function, which is defined as the integral…”

“…The preceding result provided the way to the question of the precise value of the norm of H acting on the weighted Bergman spaces. Karapetrović considered H on A p α in [12], where he derives the exact value of the norm when 4 ≤ 2(2 + α) ≤ p < ∞, that is…”

Exact essential norm of generalized Hilbert matrix operators on classical analytic function spaces

“…In recent years, a significant interest has arisen to compute the exact norm of the Hilbert matrix operator H on classical spaces of analytic functions on the open unit disk, such as Hardy spaces, weighted Bergman spaces and the Korenblum spaces, see [9], [3], [16], [17], [18] and [19]. A central tool in determining the norm of H on these spaces is an integral representation of H in terms of certain weighted composition operators established by Diamantopoulos and Siskakis in [8].…”