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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Spectra and measure inequalities

Author: C. R. Putnam
Journal: Trans. Amer. Math. Soc. 231 (1977), 519-529
MSC: Primary 47A30
MathSciNet review: 0487511
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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\Vert D \right\Vert^{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$.

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PII: S 0002-9947(1977)0487511-X
Keywords: Hilbert space, spectra of operators, hyponormal operators, analytic capacity, continuous analytic capacity, planar measure
Article copyright: © Copyright 1977 American Mathematical Society