Let T ∈ B(H ) n be an essentially normal spherical isometry with empty point spectrum on a separable complex Hilbert space H , and let A T ⊂ 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 A T has the form S = X + K with a T -Toeplitz operator X and a compact operator K. Our proof actually covers a larger class of subnormal operator tuples, called A-isometries, which includes for example the tuple T = (M z 1 , . . . , M z n ) ∈ B(H 2 (σ )) n consisting of the multiplication operators with the coordinate functions on the Hardy space H 2 (σ ) associated with the normalized surface measure σ on the boundary ∂D of a strictly pseudoconvex domain D ⊂ C n . As an application we determine the essential commutant of the set of all analytic Toeplitz operators on H 2 (σ ) and thus extend results proved by Davidson (1977) [6] for the unit disc and Ding and Sun (1997) [11] for the unit ball.

Available online xxxx Communicated by P. Biane MSC: primary 47B35 secondary 47A13, 32A35We construct a Toeplitz projection for every regular A-isometry T ∈ B(H) n on a complex Hilbert space H and use it to determine the essential commutant of the set of all analytic Toeplitz operators formed with respect to an essentially normal regular A-isometry. We show that the Toeplitz projection annihilates the compact operators if and only if T possesses no joint eigenvalues. As an application we deduce an essential version of the classical Hartman-Wintner spectral inclusion theorem, give a new proof of Johnson and Parrot's theorem on the essential commutant of abelian von Neumann algebras for separable Hilbert spaces and construct short exact sequences of Toeplitz algebras.

No abstract