A note on Toeplitz operators in Schatten-Herz classes associated with rearrangement-invariant spaces (Q1030509)

Let \(\mu\) be a positive Borel measure on the open unit ball \(B\) in \(\mathbb{R}^n\). The present paper studies Toeplitz operators \(T_\mu\) with symbol \(\mu\) acting on the harmonic Bergman space \(b^2(B)\) over \(B\). Associated to some rearrangement invariant sequence space \(E\) and generalizing the ideal of Schatten-\(p\)-class operators, one can define the Schatten ideal \(\mathcal{S}_E\) in the class of all compact operators acting on \(b^2(B)\). Under suitable conditions, it is shown that \(T_{\mu}\in \mathcal{S}_E\) is equivalent to the Berezin transfrom \(\tilde{\mu}\) or the average function \(\widehat{\mu}_{\delta}\) for small \(0<\delta <1\) of \(\mu\) being in a certain rearrangement invariant function space over \(B\). Moreover, the norms of \(T_{\mu}\), \(\tilde{\mu}\) and \(\widehat{\mu}_{\delta}\) are equivalent. In the case of the Lorentz function space \(E\), this result was proven previously by \textit{B.\,R.\thinspace Choe, H.\,Koo} and \textit{K.\,Na} [Nagoya Math.\ J.\ 185, 31--62 (2007; Zbl 1167.47022); cf.\ also ibid.\ 174, 165--186 (2004; Zbl 1067.47039)]. Given two arbitrary rearrangement invariant sequence spaces \(E\) and \(F\), the corresponding averaging-Herz class for \(\delta >0\) and the Berezin-Herz class of positive finite Borel measures on \(B\) are defined. One the level of operators, one has the notion of Schatten-Herz classes \(\mathcal{S}_E^F\). As the main theorem of the present paper, the author charaterizes the membership of a Toeplitz operator \(T_\mu\) in \(\mathcal{S}_E^F\). A necessary and sufficient condition requires \(\mu\) to be in the averaging-Herz class or in the Berezin-Herz class for small positive \(\delta\). Again, this result can be specialized to the sequence spaces \(E=\ell^p\) and \(F=\ell^q\), where \(1\leq p,q\leq \infty\). By comparing the above characterizations of \(T_{\mu}\in \mathcal{S}_{\ell^p}^{\ell^q}\) with a slightly different one due to \textit{B.\,R.\thinspace Choe, H.\,Koo} and \textit{K.\,Na} [loc.\,cit.], one obtains several equivalent definitions of the classical Herz-classes \(\mathcal{H}_p^q\).