Inner functions and spherical isometries

Abstract:

A commuting tuple $T =(T_1, \ldots , T_n) \in B(H)^n$ of bounded Hilbert-space operators is called a spherical isometry if $\sum _{i=1}^n T_i^*T_i = 1_H$. B. Prunaru initiated the study of $T$-Toeplitz operators, which he defined to be the solutions $X \in B(H)$ of the fixed-point equation $\sum _{i=1}^n T_i^*XT_i = X$. Using results of Aleksandrov on abstract inner functions, we show that $X \in B(H)$ is a $T$-Toeplitz operator precisely when $X$ satisfies $J^*XJ=X$ for every isometry $J$ in the unital dual algebra $\mathcal {A}_T \subset B(H)$ generated by $T$. As a consequence we deduce that a spherical isometry $T$ has empty point spectrum if and only if the only compact $T$-Toeplitz operator is the zero operator. Moreover, we show that if $\sigma _p(T) = \emptyset$, then an operator which commutes modulo the finite-rank operators with $\mathcal {A}_T$ is a finite-rank perturbation of a $T$-Toeplitz operator.