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Operator analogues of Mahler's measure

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.