Asymmetric Truncated Toeplitz Operators of Rank One

A truncated Toeplitz operator is a compression of the multiplication operator to a backward shift invariant subspace of the Hardy space H-2. An asymmetric truncated Toeplitz operator is a compression of the multiplication operator that acts between two different backward shift invariant subspaces of H-2. All rank-one truncated Toeplitz operators have been described by Sarason. Here, we characterize all rank-one asymmetric truncated Toeplitz operators. This completes the description given by Lanucha for asymmetric truncated Toeplitz operators on finite-dimensional backward shift invariant subspaces.