Schatten class composition operators on weighted Bergman spaces of bounded symmetric domains
Let \(\Omega\) be a bounded symmetric domain in complex Euclidean \(n\)-dimensional space \(\mathbb{C}^n\), and if \(H\) with an inner product is a Hilbert space and \(0<p<\infty\), let \(S_p(H)\) be the Schatten \(p\)-class of compact operators on \(H\). The Bergman space \(A^2(\Omega)\) is the subspace of \(L^2(\Omega)\) consisting of holomorphic functions. If \(K\) is the Bergman kernel associated with the Bergman projection \(P: L^2(\Omega)\to A^2(\Omega)\), then \(K(z,w)= h(z,w)^{-N}\), where \(h(z,w)= h_\Omega(z,w)\) is a polynomial and \(N= N_\Omega\) is a positive integer. In addition, if \(v\) is the Lebesgue measure on \(\mathbb{R}^{2n}\), then the measure \(v^a\) by \(v^a(A)= \int_A c^{-1}_a K(z,z)^a dv(z)\), \(c_a= \int_\Omega K(z,z)^a dv(z)\). The composition operator \(C_\phi\) is defined in terms of a holomorphic function \(\phi: \Omega\to\Omega\) by \(C_\phi(f)(z)= f\circ \phi(z)= f(\phi(z))\). If \(0<p<\infty\), \(E(z,r)= \{w\in\Omega: \beta(z,w)< r\}\), where \(\beta\) is the Bergman metric, and if \(b^a_\phi= v^a(\phi^{-1}(E(z,r)))|E(z,r)|^{a- 1}\), then the main theorem of this paper shows that, with \(A_a\) associated with \(v^a\), (i) the compact operator \(C_\phi\) is in \(S_{2p}(A^2_a(\Omega))\) if and only if \(b^a_\phi\in L^p(\Omega, d\lambda)\), where \(d\lambda(z)= c_aK(z,z)^{1- a}dv^a(z)\); (ii) for \(2(1- a_\Omega)/(1- a)< p<\infty\), \(a_\Omega= N^{-1}_\Omega\), the compact operator \(C_\phi\) is in \(S_p(A^2_a(\Omega))\) if and only if \(B^a_\phi\in L^p(\Omega,d\lambda)\), where \(B^a_\phi(z)^2= K(z,z)^{a- 1}\int_\Omega|K(z,\phi(w))|^{2- 2a} dv^a(w)\).
Schatten \(p\)-class of compact operators
scientific article; zbMATH DE number 1383202
scientific article