Near invariance and kernels of Toeplitz operators

This paper makes a systematic study of kernels of Toeplitz operators on scalar and vector-valued H p spaces (for 1 < p < ∞). The property of near invariance of a kernel for the backward shift is analysed and shown to hold in increased generality. In the scalar case, and in some vectorial cases, the existence of a minimal kernel containing a given function is established, and a symbol for a corresponding Toeplitz operator is determined; thus, for rational symbols, its dimension can be easily calculated. It is shown that every Toeplitz kernel in H p is the minimal kernel for some function lying in it.