On the essential commutant of 𝒯(𝒬𝒞)

Let T {\mathcal T} (QC) (resp. T {\mathcal T} ) be the C ∗ C^\ast -algebra generated by the Toeplitz operators { T φ : φ ∈ \{T_\varphi : \varphi \in QC } \} (resp. { T φ : φ ∈ L ∞ } \{T_\varphi : \varphi \in L^\infty \} ) on the Hardy space H 2 H^2 of the unit circle. A well-known theorem of Davidson asserts that T {\mathcal T} (QC) is the essential commutant of T {\mathcal T} . We show that the essential commutant of T {\mathcal T} (QC) is strictly larger than T {\mathcal T} . Thus the image of T {\mathcal 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 {\mathcal T} (QC).