  We consider kernels of unbounded Toeplitz operators in $H^p(\mathbb C^+)$ in
terms of a factorization of their symbols. We study the existence of a minimal
Toeplitz kernel containing a given function in $H^p(\mathbb C^+)$, we describe
the kernels of Toeplitz operators whose symbol possesses a certain
factorization involving two different Hardy spaces and we establish relations
between the kernels of two operators whose symbols differ by a factor which
corresponds, in the unit circle, to a non-integer power of $z$. We apply the
results to describe the kernels of Toeplitz operators with non-vanishing
piecewise continuous symbols.
