Dual Toeplitz operators on the sphere via spherical isometries
Let \(\mathbb{B}_n\) be the open Euclidean unit ball in \(\mathbb{C}^n\), \(\mathbb{S}_n=\partial\mathbb{B}_n\) its boundary and \(\sigma\) the normalized surface measure on \(\mathbb{S}_n\). Given \(\varphi\in L^\infty(\sigma)\), the Toeplitz operator \(T_\varphi\) with symbol \(\varphi\) is defined as the compression of the multiplication operator \(M_\varphi: L^2(\sigma)\to L^2(\sigma),\;\;f\mapsto\varphi f\), onto the Hardy space \(H^2(\sigma)\), and the dual Toeplitz operator on the space \(H^2(\sigma)^\perp\) is given by \[ S_\varphi:\;H^2(\sigma)^\perp\to H^2(\sigma)^\perp,\;\;f\mapsto P_{H^2(\sigma)^\perp}M_\varphi f. \] For a separable complex Hilbert space \(H\), a spherical isometry is a commuting tuple \(T=(T_1,\dots,T_n)\in B(H)^n\) of bounded linear operators on \(H\) satisfying the algebraic condition \(\sum_{i=1}^n T_i^*T_i=1_H\). An operator \(X\in B(H)\) is called a Toeplitz operator with respect to a given spherical isometry \(T\in B(H)^n\) (\(T\)-Toeplitz operator, for short) if the identity \(\sum_{i=1}^n T_i^*XT_i=X\) holds. It is shown that the tuple \(T=(S_{\overline{z}_1},\dots,S_{\overline{z}_n})\in B(H^2(\sigma)^\perp)^n\) is a spherical isometry, and the dual Toeplitz operators on \(H^2(\sigma)^\perp\) are precisely the associated \(T\)-Toeplitz operators. A short exact sequence for the dual Toeplitz \(C^*\)-algebra \(C^*(S_f: f\in C(\mathbb{S}_n))\) is established. A characterization of dual Toeplitz operators using automorphisms of the unit ball is obtained. The commutator ideal of the dual Toeplitz \(C^*\)-algebra is determined.
scientific article; zbMATH DE number 6499543
scientific article