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David Békollè

Publications and source records attributed to David Békollè.

4 recordsLinked to original sources
arXiv2026

Weakly, sufficiently or strongly localized operators on the Fock space in \mathh C^n

We study properties of the following four classes of operators on the Fock space in $\mathbb C^n:$ 1) weakly localized operators; 2) sufficiently localized operators in the sense of Xia and Zheng; 3) sufficiently localized operators; 4) strongly localized operators. In this respect, we examine composition operators, Toeplitz operators with a measure symbol whose total variation measure is a Fock-Carleson measure, and singular operators of convolution type introduced by Zhu, among others. We also provide a bounded operator which is not weakly localized and does not even belong to the Toeplitz algebra. Class 1) contains class 2), class 2) contains class 3), which clearly contains class 4). We prove that the first two inclusions are strict. Our proofs are in terms of singular operators of convolution type introduced by Zhu. The third inclusion was already known to be strict, as Wang, Cao and Zhu exhibited examples of composition operators which are sufficiently localized, but are not strongly localized. %As our main result, we show the existence of a singular operator of convolution type which is weakly localized, but is not sufficiently localized in the sense of Xia and Zheng. %The underlying question is whether the first two classes of operators coincide or not.

math.FA↗
arXiv2025

Analysis of Toeplitz Operators with $BMO^1_α$ operator-valued symbols on $\ell^2-$Valued Bergman Spaces

As a class of compact operators on the $\ell^2-$valued Bergman space $A^2_α(\mathbb B_n, \ell^2)$ on the unit ball $\mathbb B_n,$ we study Toeplitz operators with $BMO^1_α(\mathbb B_n, \mathcal L(\ell^2))$ operator-valued symbols. First, we describe a method of restriction to a finite dimension which allows us to apply earlier results of Rahm and Wick; then we exhibit an explicit example of a compact Toeplitz operator on $A^2_α(\mathbb B_n, \ell^2).$ Secondly, we apply two sufficient conditions for compactness established by Rahm in infinite dimension. The first condition is in terms of the Toeplitz algebra $\mathcal T_{L^\infty_{fin}},$ the second one is in terms of sufficiently localized operators and is implied by the first condition. To get the second condition, we additionally assume that the symbol and its adjoint belong to $BMO^1_α(\mathbb B_n, \mathcal L(\ell^2, \ell^1)).$ Finally, inspired by Xia and Sadeghi-Zorboska, we ask the question of the validity of the reverse implication.

math.CA↗
arXiv2020

About a conjecture of Lieb-Solovej

Very recently, E. H. Lieb and J. P. Solovej stated a conjecture about the constant of embedding between two Bergman spaces of the upper-half plane. A question in relation with a Werhl-type entropy inequality for the affine $AX+B$ group. More precisely, that for any holomorphic function $F$ on the upper-half plane $Π^+$, $$\int_{Π^+}|F(x+iy)|^{2s}y^{2s-2}dxdy\le \frac{π^{1-s}}{(2s-1)2^{2s-2}}\left(\int_{Π^+}|F(x+iy)|^2 dxdy\right)^s $$ for $s\ge 1$, and the constant $\frac{π^{1-s}}{(2s-1)2^{2s-2}}$ is sharp. We prove differently that the above holds whenever $s$ is an integer and we prove that it holds when $s\rightarrow\infty$. We also prove that when restricted to powers of the Bergman kernel, the conjecture holds. We next study the case where $s$ is close to $1.$ Hereafter, we transfer the conjecture to the unit disc where we show that the conjecture holds when restricted to analytic monomials. Finally, we overview the bounds we obtain in our attempts to prove the conjecture.

math.CV↗