Aluthge Transforms and the Convex Hull of the Spectrum of a Hilbert Space Operator

Abstract

AbstractFor a bounded linear operator T on a Hilbert space its Aluthge transform Δ(T) is defined as Δ (T) = |T|1/2U|T|1/2 with the help of a polar representation T = U|T|. In recent years usefulness of the Aluthge transform has been shown in several directions. In this paper we will use the Aluthge transform to study when the closure of the numerical range W(T) of T coincides with the convex hull of its spectrum. In fact, we will prove that it is the case if and only if the closure of W(T) coincides with that of W(Δ(T)). As a consequence we will show also that for any operator T the convex hull of its spectrum is written as the intersection of the closures of the numerical ranges of all iterated Aluthge transforms Δn(T).KeywordsAluthge transformNumerical rangeConvex hull of spectrumConvexoid operatorNorm inequalities