Compactness for Toeplitz and Hankel operators on weighted Bergman spaces of the ball in \(\mathbb{C}^ n\)
Let \(B_ n= \{z\in \mathbb{C}^ n;\;\| z\|< 1\}\) be the open unit ball, \(H(B_ n)\) the set of holomorphic functions on \(B_ n\), \(\nu\) the normalized Lebesgue measure on \(B_ n\), and \[ d\nu_ \alpha(z)= [\Gamma(n+ \alpha+ 1)/n!\Gamma(\alpha+1)] (1-\| z\|^ 2)^ \alpha d\nu(z),\quad \alpha>-1. \] Let \(L^{2,\alpha}(B_ n)\) be the space of measurable functions \(f: B_ n\to \mathbb{C}\) such that \[ \| f\|_{2,n,\alpha}= \left[\int_{B_ n} | f(z)|^ 2 d\nu_ \alpha(z)\right]^{1/2}<\infty; \] and the Bergman space \(A^{2,\alpha}(B_ n)= L^{2,\alpha}(B_ n)\cap H(B_ n)\). Let \(P^{n,\alpha}: L^{2,\alpha}(B_ n)\to A^{2,\alpha}(B_ n)\) be the projection defined by: \[ (P^{n,\alpha} f)(w)= \int_{B_ n} \bigl[f(z)/\overline{(1-\langle z,w\rangle)}^{n+\alpha+1}\bigr] d\nu_ \alpha(z); \] and let \(\phi_ w: B_ n\to B_ n\) be the Möbius map determined by \(w\in B_ n\): \[ \phi_ w(z)= \bigl[w- P_ w z- (1-\| w\|^ 2)^{1/2} Q_ w z\bigr]/[1-\langle z,w\rangle],\quad z\in B_ n. \] For \(f\in L^ \infty(B_ n)\), let \(T^{(n,\alpha)}_ f\), \(H^{(n,\alpha)}_ f\) be the Toeplitz and Hankel operators defined by: \[ \begin{aligned} (T^{(n,\alpha)}_ f g)(w) &= (P^{n,\alpha} fg)(w),\quad g\in A^{2,\alpha}(B_ n);\\ (H^{(n,\alpha)}_ f g)(w) &= ((I- P^{n,\alpha})(fg))(w),\quad g\in A^{2,\alpha}(B_ n).\end{aligned} \] Then the author proves that: \(T^{(n,\alpha)}_ f\) is compact on \(A^{2,\alpha}(B_ n)\) if and only if \[ \| (P^{n,\alpha}(f(\phi_ w))\|_{2,n,\alpha}\underset{\| w\|\to 1}\longrightarrow 0; \] \(H^{(n,\alpha)}_ f\) is compact on \(A^{2,\alpha}(B_ n)\) if and only if \[ \|(I- P^{n,\alpha})(f(\phi_ w))\|_{2,n,\alpha}\underset{\| w\|\to 1} \longrightarrow 0; \] solving an open problem of S. Axler.
scientific article; zbMATH DE number 495927