Generalized uncertainty inequalities

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In this paper, Heisenberg-Pauli-Weyl-type uncertainty inequalities are obtained for a pair of positive-self adjoint operators on a Hilbert space, whose spectral projectors satisfy a ``balance condition'' involving certain operator norms. This result is then applied to obtain uncertainty inequalities on Riemannian manifolds, Riemannian symmetric spaces of non-compact type, homogeneous graphs and unimodular Lie groups with sublaplacians.