The commutant and invariant subspaces for dual truncated Toeplitz operators

Abstract

Dual truncated Toeplitz operators on the orthogonal complement of the model space $$K_u^2(=H^2 \ominus uH^2)$$ with u nonconstant inner function are defined to be the compression of multiplication operators to the orthogonal complement of $$K_u^2$$ in $$L^2$$ . In this paper, we give a complete characterization of the commutant of dual truncated Toeplitz operator $$D_z$$ , and we even obtain the commutant of all dual truncated Toeplitz operators with bounded analytic symbols. Moreover, we describe the nontrival invariant subspaces of $$D_z$$ .