\(m\)-Berezin transform and compact operators
The authors introduce and obtain some basic properties of the \(m\)-Berezin transforms of bounded operators on the Bergman space of the unit ball \(B\) in \(\mathbb C^n\). The Bergman space \(L^2_a\) is the space of analytic functions on \(B\) which are square integrable with respect to the Lebesgue volume measure \(d\nu\). Let \(\phi_z\) denote the automorphism of \(B\) that interchanges \(0\) and \(z\). For a nonnegative integer \(m\), the \(m\)-Berezin transform of an integrable function \(f\) can be defined by \[ B_mf(z)=\int_Bf\circ\phi_z(u)\,d\nu_m(u) \] for \(z\in B\), where \(d\nu_m(u)= \binom{m+n}{n} (1-|u|^2)^m\,d\nu(u)\). When \(m=0\), this is just the usual Berezin transform. \textit{P. Ahern, M. Flores} and \textit{W. Rudin} studied these transforms and obtained some useful properties in [J. Funct. Anal. 111, No. 2, 380--397 (1993; Zbl 0771.32006)]. In the paper under review, the authors appropriately generalize \(m\)-Berezin transforms to operators and show some important properties of these transforms. For example, these transforms commute with each other. In other words, for a bounded operator \(S\) on \(L^2_a\) and \(j,m\geq 0\), \((B_mB_j)(S)=(B_jB_m)(S)\). The authors also compute the norm of \(B_m\). For \(z\in B\), denote by \(U_z\) the unitary operator \(U_zf=(f\circ\phi_z)(-1)^nk_z\), where \(k_z\) denotes the normalized Bergman kernel for \(B\). For a bounded linear operator \(S\), set \(S_z=U_zSU_z\). The authors show that \(B_m(S_z)=(B_mS)\circ\phi_z\) for any \(m\geq 0\) and \(z\in B\). They also generalize a result of \textit{L. A. Coburn} [Proc. Am. Math. Soc. 133, No. 1, 127--131 (2005; Zbl 1093.47024)] and show that \(B_mS(z)\) is Lipschitz with respect to the pseudo-hyperbolic distance on \(B\). Finally, using the \(m\)-Berezin transform it is shown that a radial operator in the Toeplitz algebra is compact if and only if its Berezin transform vanishes on the boundary of the unit ball. Recently, D. Suárez announced that he proved this result for any operator in the Toeplitz algebra.
scientific article; zbMATH DE number 5149122