Finite-rank and compact defect operators of truncated Toeplitz operators

Abstract

In this paper, we provide the necessary and sufficient condition that the defect operator I−Aφ⁎Aφ of truncated Toeplitz operator Aφ for φ∈Ku2∩L∞ with ‖φ‖∞≤1 is of finite-rank on the model space Ku2. Moreover, the necessary and sufficient condition is obtained for I−Aφ⁎Aφ with φ∈Ku2∩L∞ to be compact on the model space Ku2.