Norm estimates of weighted composition operators pertaining to the Hilbert matrix

Abstract

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.

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

“…The inequality B 2 p , 2p − 4 ≤ 1 (p−2)(4−p) holds for 2 < p < 4, by Lemma 2.5 in [4] or Lemma 3.2 in [13]. The above inequality is one ingredient in the proof of the main result in [4].…”

“…Božin and Karapetrović [4] confirmed the conjecture in the positive by reducing the problem to certain novel estimates of the Beta function. In [13] the authors simplified the proofs of the key lemmas in [4] significantly by discarding the use of a classical theorem of Sturm.…”

“…In this article, which is a continuation of [13], the results concerning the unweighted Bergman space case are generalized to the weighted Bergman spaces A p α , where α ≥ 0. The study of the boundedness of the Hilbert matrix operator H acting on A p α was initiated in the article [8] where partial results were obtained.…”

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].…”

“…, which is an improvement. Regarding the Korenblum spaces H ∞ α , 0 < α < 1, the exact norm of H was computed for small values of α and an upper estimate was established for large values of α in [18]. See also [1].…”