Compact commutators of truncated Toeplitz operators on the model space

Abstract

Let u be an inner function and $K_u$ be the corresponding model space. In this article, we characterize the compact commutator $[A_f,\,A_g]:=A_fA_g-A_gA_f$ of truncated Toeplitz operators $A_f$ and $A_g$ on $K_u.$ For f and g inner functions, we discuss the relationship between closed singular set $\sigma (u)$ and the compactness of $[A_{f},\,A_{g}^*].$ In particular, when $u=fu_1$ for inner function $u_1,$ we present necessary and sufficient conditions for $[A_f,\,A_f^*]$ to have finite rank or to be compact on $K_u.$ For $f,\,g\in H^\infty ,$ in view of symbols of truncated Toeplitz operators, we show a sufficient condition for $[A_f,\,A_g^*]$ to be compact. For $f,\,g\in L^\infty ,$ we obtain necessary and sufficient conditions for $[A_f,\,A_g]$ to be compact using the function algebra.