On the essential commutant of T (QC)

Let T (QC) (resp. T ) be the C*-algebra generated by the Toeplitz operators {Tℓ : ℓ ε QC} (resp. {Tℓ : ℓ ε L ^∞}) on the Hardy space H² of the unit circle. A well-known theorem of Davidson asserts that T (QC) is the essential commutant of T . We show that the essential commutant of T (QC) is strictly larger than T . Thus the image of T in the Calkin algebra does not satisfy the double commutant relation. We also give a criterion for membership in the essential commutant of T (QC).