On the essential commutant of analytic Toeplitz operators associated with spherical isometries

Let T ∈ B(H)n be an essentially normal spherical isometry with empty point spectrum on a separable complex Hilbert space H, and let AT ⊂ B(H) be the unital dual operator algebra generated by T . In this note we show that every operator S ∈ B(H) in the essential commutant of AT has the form S = X+K with a T -Toeplitz operator X and a compact operatorK. Our proof actually covers a larger class of subnormal operator tuples, called A-isometries, which includes for example the tuple T = (Mz1, . . . , Mzn ) ∈ B(H2(σ ))n consisting of the multiplication operators with the coordinate functions on the Hardy space H2(σ ) associated with the normalized surface measure σ on the boundary ∂D of a strictly pseudoconvex domain D ⊂ Cn. As an application we determine the essential commutant of the set of all analytic Toeplitz operators on H2(σ ) and thus extend results proved by Davidson (1977) [6] for the unit disc and Ding and Sun (1997) [11] for the unit ball. © 2011 Elsevier Inc. All rights reserved