Kernels of Toeplitz operators on the Hardy space over the bidisk

We study the kernels of Toeplitz operators on the Hardy space on the bidisk. We first give a sufficient condition for a general symbol to be antiholomorphic under the assumption that the kernel of the corresponding Toeplitz operator contains a backward shift invariant subspace. As an application, we construct an invariant subspace whose its orthogonal complement can not be the kernel of any Toeplitz operator. Also, we give a characterization on a Toeplitz operator for which the orthogonal complement of its kernel is generated by certain inner functions. Finally, we describe all backward shift invariant subspaces which are in the kernels of Toeplitz operators with homogeneous type symbols. As an application, we show that there is a Toeplitz operator for which its kernel has dimension of any given integer and the orthogonal complement of the kernel is generated by two functions. Our result shows that there are higher dimensional phenomena.