On numerical ranges and elementary operators

Abstract

The numerical range of a bounded linear operator on a Hilbert space H, is the range of the restriction to the unit sphere of the quadratic form associated with the operator. An elementary operator is a bounded linear mapping on the set of bounded linear operators acting on an infinite dimensional complex Hilbert space. Properties of elementary operators have been investigated during the past three decades under a variety of aspects such as their spectra, compactness, norm properties, numerical range among others. However, through all these studies it emerges that, for a general elementary operator, a precise description of its prop- erties has not been explored exhaustively. Thus a generalized description of these properties on the various generalizations of numerical ranges of an elementary op- erator is missing and hence have been studied in this research work. The general objective was to establish the relations that exists between the numerical range of the elementary operator and that of the implementing operators as the operator acts on the various algebras. Specifically, the objectives were: to establish some of the properties of the numerical range that hold for an elementary operator acting the algebra L(H) and to determine the relationship that exists between the ele- mentary operators and numerical ranges of their implementing operators both in a normed ideal and in a Hilbert space considered as a C ∗ −algebra. In particular, the convexity of the algebra numerical range was shown as well as its equality to the algebra numerical range of the left and right multiplication operators. The algebra numerical range of a generalized derivation restricted to a norm ideal was established to be equal to the set difference of the algebra numerical ranges of the implementing operators. Finally, the closed convex hull of the maximal numerical range of the implementing operators was shown to be contained in the algebraic maximal numerical range of an elementary operator restricted to an operator al- gebra. Working from the known to the unknown, we have borrowed from the already established relationships between the spectrum of an elementary operator and the joint spectrum of two commuting n-tuples and obtained relations in terms of the numerical ranges. Another approach utilized was algebraically constructive in nature. From the theory of Banach spaces, we have for instance, the famous Hahn Banach theorem that allows us to algebraically construct functionals in a subspace and we are guaranteed of an extension in the whole space under consider- ation. With regards to the application of our findings, the numerical range is often used to locate the spectrum of an operator. Certain problems in quantum me- chanics, for instance, approximation by commutators, the Heisenberg uncertainly principle, among others correspond to elementary operators and the findings ob- tained from our research will contribute to the theoretical knowledge that such physicists and applied mathematicians need.