Spectra and measure inequalities

Abstract:

Let T be a bounded operator on a Hilbert space $\mathfrak {H}$ and let ${T_z} = T - zI$. Then the operators ${T_z}T_z^\ast ,{T_z}{T_t}{({T_z}{T_t})^\ast }$, and ${T_z}{T_t}{T_s}{({T_z}{T_t}{T_s})^\ast }$ 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 $\mathfrak {H}$ be separable and suppose that ${T_z}{T_t}{({T_z}{T_t})^\ast } \geqslant D \geqslant 0$ 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 ${\left \| D \right \|^{1/2}} \leqslant {\pi ^{ - 1}}\;{\text {meas}_2}\;({\sigma _p}(T))$. If T is the adjoint of the simple unilateral shift, then equality holds with $D = I - {T^\ast }T$.