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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. |
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