  Compressions of Toeplitz operators to coinvariant subspaces of $H^2$ are
called truncated Toeplitz operators. We study two questions related to these
operators. The first, raised by Sarason, is whether boundedness of the operator
implies the existence of a bounded symbol; the second is the reproducing kernel
thesis. We show that in general the answer to the first question is negative,
and we exhibit some classes of spaces for which the answers to both questions
are positive.
