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Abstract
Let denote the rank-
numerical range of an
-by-
complex matrix
. We give a characterization for
, where
, via the compressions and the principal submatrices of
. As an application, the matrix
satisfying
, where
is the classical numerical range of
and
, is under consideration. We show that if
for some
, then
is unitarily similar to
, where
is a 2-by-2 matrix,
is a
-by-
matrix and
.
AMS Subject Classification:
Acknowledgments
We appreciate the advice from Professor Pei Yuan Wu. He pointed out that there exists an 6-by-6 unitarily irreducible matrix such that
and therefore, the number
in Theorem 2.9 (c) is best possible. We also thank the referee for his/her comments, which improved both the statement and the proof of Theorem 2.2. The Research suppported by the National Science Council of the Republic of China under the projects NSC 101-2115-M-035-006, NSC 101-2115-M-008-006 and NSC 101- 2115-M-009-001, respectively.