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

Abstract

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.

Toeplitz projections and essential commutants

“…Thus we improve a corresponding result obtained in [11] under the additional condition that T possesses no joint eigenvalues. We obtain complete characterizations of the essential commutant of essentially normal regular A-isometries in Section 5 and give, as a direct application, a new proof of a classical theorem of Johnson and Parrot [16] on the essential commutant of abelian von Neumann algebras in the case of separable Hilbert spaces.…”

“…By Lemma 3.9 (c) and Lemma 4.1 in [11], the essential normality of T yields that D = (T ) ec is a C * -algebra containing T (T ) ∪ K(H), that the C * -algebra…”

“…In [11] the above results were extended to Toeplitz operators formed with respect to a quite general class of subnormal tuples on arbitrary Hilbert spaces containing, as a very particular case, Toeplitz operators on strictly pseudoconvex domains in C n .…”

“…Under a suitable regularity condition on T , which is satisfied in all the above examples and which is needed to apply results of Aleksandrov [1] on the existence of sufficiently many μ-inner functions, it follows that the set T(T ) of abstract T -Toeplitz operators is given by the compressions It follows from results of B. Prunaru [19] on families of spherical isometries that there is a completely positive unital projection Φ T : B(H) → B(H) onto the set T(T ) of all abstract T -Toeplitz operators [11]. In this note we give a much more direct and straightforward construction of Toeplitz projections Φ T .…”

“…In this note we give a much more direct and straightforward construction of Toeplitz projections Φ T . We use the properties of these projections to improve the main result of [11] on the essential commutant of analytic Toeplitz operators and to extend a number of classical results on Toeplitz operators to our general setting.…”