Estimating Norms of Commutators
We find estimates on the norm of a commutator of the form [f(x), y] in terms of the norm of [x, y], assuming that x and y are bounded linear operators on Hilbert space, with x normal and with spectrum within the domain of f. In particular, we discuss ‖[x2, y]‖ and ‖[x1/2, y]‖ for 0 ≤ x ≤ 1. For larger values of δ = ‖[x, y]‖, we can rigorously calculate the best possible upper bound ‖[f(x), y]‖ ≤ ηf(δ) for many f. In other cases, we have conducted numerical experiments that strongly suggest that we have in many cases found the correct formula for the best upper bound.
Norm Estimates for Commutators of Operators
Given two self‐adjoint Hilbert space operators A, B, and a continuous function f, we prove several inequalities of the form
Axonal sensorimotor neuropathy in patients with β-thalassaemia
Objective: The purpose of this study was to investigate the prevalence of peripheral neuropathy in patients with β-thalassaemia. Methods: Thirty six patients with a mean age of 29.2±8.2 years and 17 healthy controls with a mean age of 27.6±9.1 were included in this study. Measurements included the neuropathy symptoms score (NSS), the neuropathy disability score (NDS) as well as nerve conduction studies of two motor (ulnar and peroneal) and two sensory (ulnar and sural) nerves of the right limbs. Results: A mainly sensory axonal polyneuropathy was present in 19 out of 36 patients (52.7%). Eight out of these 19 patients also had abnormal NDS values. The neuropathy correlated significantly with the age of the patients and the hematocrit. However, it did not correlate with the presence of antibodies against HCV, the ferritin levels, or with a history of transfusions, desferrioxamine treatment, or splenectomy. Conclusions: This study showed a high prevalence of a predominantly sensory neuropathy in patients with β-thalassaemia. The electrophysiological data suggest that the underlying pathology is an axonopathy. Chronic hypoxia of the nerves resulting from severe anaemia may contribute to the pathogenesis of this neuropathy.