Inner functions and spherical isometries

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