  In this paper we study the kernels of Toeplitz operators on both the scalar
and the vector-valued Hardy space for $ 1 < p < \infty $. We show existence of
a minimal kernel of any element of the vector-valued Hardy space and we
determine a symbol for the corresponding Toeplitz operator. In the scalar case
we give an explicit description of a maximal function for a given Toeplitz
kernel. In the vectorial case we show not all Toeplitz kernels have a maximal
function and in the case of $p=2$ we find the exact conditions for when a
Toeplitz kernel has a maximal function. For both the scalar and vector-valued
Hardy space we study the minimal Toeplitz kernel containing multiple elements
of the Hardy space, we also find an equivalent condition for a function in the
Smirnov class to be cyclic for the backwards shift.
