  Motivated by a geometric meaning of Mahler's measure, we introduce two
operator analogues of Mahler's measure. This leads to some interesting
equalities and inequalities between the two operator-theoretic Mahler measures
and the classical Mahler measure. In order to apply these results to the
operator version of Lehmer's problem, we introduce and study an important class
of operators, the so-called subharmonic operators. It is shown that the
operator version of Lehmer's problem fails under some mild condition.
