  The geometric descriptions of the (essential) spectra of Toeplitz operators
with piecewise continuous symbols are among the most beautiful results about
Toeplitz operators on Hardy spaces $H^p$ with $1<p<\infty$. In the Hardy space
$H^1$, the essential spectra of Toeplitz operators are known for continuous
symbols and symbols in the Douglas algebra $C+H^\infty$. It is natural to ask
whether the theory for piecewise continuous symbols can also be extended to
$H^1$. We answer this question in negative and show in particular that the
Toeplitz operator is never bounded on $H^1$ if its symbol has a jump
discontinuity.
