Properties of Toeplitz operators on some holomorphic Banach function spaces
For \(\alpha>-1\) and a positive Borel measure \(\mu\) on the unit ball \({\mathbb B}_{n}\subset {\mathbb C}_{n}\) such that NEWLINE\[NEWLINE \left| \int_{{\mathbb B}_{n}} (1-|w|^{\alpha}) \,d\mu(w) \right| <\infty, NEWLINE\]NEWLINE the associated Toeplitz operator is defined by the formula NEWLINE\[NEWLINE (T_{\mu}^{\alpha}f)(z) = c_{\alpha} \int_{{\mathbb B}_{n}} \frac{(1-|w|^{2})^{\alpha}f(w)}{(1-\langle z,w \rangle)^{n+\alpha+1}} \, d\mu(w), NEWLINE\]NEWLINE where \(w\in{\mathbb B}_{n}\), \(\langle \cdot,\cdot \rangle\) is the scalar product in \({\mathbb C}_{n}\), \(c_{\alpha}\) is the normalizing constant.NEWLINENEWLINEUnder some additional assumptions, the authors characterize those measures for which these operators are bounded or compact on some spaces of holomorphic functions on \({\mathbb B}_{n}\), including the case of Besov spaces.
scientific article; zbMATH DE number 6063096
scientific article