J. Aust. Math. Soc.
82 (2007), 325343

Numerical range of the derivation of an induced operator

Randall R. Holmes
Department of Mathematics and Statistics
Auburn University
Auburn
Alabama 368495310
USA
holmerr@auburn.edu


ChiKwon Li
Department of Mathematics
College of William and Mary
PO Box 8795, Williamsburg
Virginia 231878795
USA
ckli@math.wm.edu



TinYau Tam
Department of Mathematics and Statistics
Auburn University
Auburn
Alabama 368495310
tamtiny@auburn.edu



Abstract

Let
V
be an
ndimensional inner product space over
, let
H
be a subgroup of the symmetric group on
, and let
be an irreducible character. Denote by
the symmetry class of tensors over
V
associated with
H
and
. Let
be the operator induced by
, and let
be the derivation operator of
T. The decomposable numerical range
of
is a subset of the classical numerical range
of
. It is shown that there is a closed starshaped
subset
of complex numbers such that 

where
denotes the convex hull of
. In many cases, the set
is convex, and thus the set inclusions are
actually equalities. Some consequences of the
results and related topics are discussed.

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