The Bergman space, the Bloch space, and commutators of multiplication operators
Let \(L^ 2_ a\) denote the Bergman space of functions analytic in the unit disc \({\mathbb{D}}\) and square summable with respect to the area measure. For \(f\in L^{\infty}({\mathbb{D}})\) the Hankel operator \(H_ f\) on \(L^ 2_ a\) is defined by \[ (1)\quad H_ f(g)=(I-{\mathbb{P}})(fg), \] where \({\mathbb{P}}\) is the orthogonal projection from \(L^ 2({\mathbb{D}})\) onto \(L^ 2_ a.\) The main result of the paper shows that for \(f\in H^{\infty}({\mathbb{D}})\) the norm of \(H_{\bar f}\) is equivalent to the fact that f belongs to the Bloch space \(\beta\). In order that the operator \(H_ f\) defined by (1), for \(\bar f\) analytic in \({\mathbb{D}}\), be a bounded operator on \(L^ 2_ a\) it is necessary and sufficient that \(\bar f\) be in \(\beta\). It is also proved that for antianalytic f the operator \(H_ f\) is compact if and only if \(\bar f\) belongs to the closure of polynomials in \(\beta\).
scientific article; zbMATH DE number 4030297
scientific article