Generalized Crofoot transform and applications

Abstract Matrix-valued asymmetric truncated Toeplitz operators are compressions of multiplication operators acting between two model spaces. These are the generalization of matrix-valued truncated Toeplitz operators. In this article, we describe symbols of matrix-valued asymmetric truncated Toeplitz operators equal to the zero operator. We also use generalized Crofoot transform to find a connection between the symbols of matrix-valued asymmetric truncated Toeplitz operators T ( Θ 1 , Θ 2 ) {\mathcal{T}}\left({\Theta }_{1},{\Theta }_{2}) and T ( Θ 1 ′ , Θ 2 ′ ) {\mathcal{T}}\left({\Theta }_{1}^{^{\prime} },{\Theta }_{2}^{^{\prime} }) .