Quasi-radial Operators on the Weighted Bergman Space over the Unit Ball

Abstract

We study the so-called quasi-radial operators, i.e., the operators that are invariant under the subgroup of the unitary group ${\mathfrak U}(n)$ formed by the block-diagonal matrices with unitary blocks of fixed dimensions. The quasi-radial Toeplitz operators appear naturally and play a crucial role under the study of the commutative Banach (not $C^*$) algebras of Toeplitz operators [1, 8]. They form an intermediate class of operators between the Toeplitz operators with radial $a=a(r)$, $r=\sqrt{|z_1|^2 + \ldots + |z_n|^2}$, and separately-radial $a = a(|z_1|, \ldots, |z_n|)$ symbols.