Kernels of Unbounded Toeplitz Operators and Factorization of Symbols

We consider kernels of unbounded Toeplitz operators in $$H^p({\mathbb {C}}^{+})$$ H p ( C + ) in terms of a factorization of their symbols. We study the existence of a minimal Toeplitz kernel containing a given function in $$H^p({\mathbb {C}}^{+})$$ H p ( 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.