Let T be a bounded operator on a Hilbert space H \mathfrak {H} and let T z = T − z I {T_z} = T - zI . Then the operators T z T z ∗ , T z T t ( T z T t ) ∗ {T_z}T_z^\ast ,{T_z}{T_t}{({T_z}{T_t})^\ast } , and T z T t T s ( T z T t T s ) ∗ {T_z}{T_t}{T_s}{({T_z}{T_t}{T_s})^\ast } are nonnegative for all complex numbers z, t, and s. We shall obtain some norm estimates for nonnegative lower bounds of these operators, when z, t, and s are restricted to certain sets, in terms of certain capacities or area measures involving the spectrum and point spectrum of T. A typical such estimate is the following special case of Theorem 4 below: Let H \mathfrak {H} be separable and suppose that T z T t ( T z T t ) ∗ ⩾ D ⩾ 0 {T_z}{T_t}{({T_z}{T_t})^\ast } \geqslant D \geqslant 0 for all z and t not belonging to the closure of the interior of the point spectrum of T. In addition, suppose that the boundary of the interior of the point spectrum of T has Lebesgue planar measure 0. Then ‖ D ‖ 1 / 2 ⩽ π − 1 meas 2 ( σ p ( T ) ) {\left \| D \right \|^{1/2}} \leqslant {\pi ^{ - 1}}\;{\text {meas}_2}\;({\sigma _p}(T)) . If T is the adjoint of the simple unilateral shift, then equality holds with D = I − T ∗ T D = I - {T^\ast }T .