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In the literature it is known that the decomposable numerical range of
is not necessarily convex. But it is not known whether
is star-shaped. We construct a symmetric unitary matrix
such that the decomposable numerical range
is not star-shaped and hence not simply connected. We then consider a real analog
and show that
is star-shaped if
is skew symmetric. Such a star-shapedness result is also true for the Pfaffian numerical range
.
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