When is the numerical range of a nilpotent matrix circular? (English) Zbl 1191.15018
The problem formulated in the title is examined. A \(2\times 2\) nilpotent matrix always has circular numerical range. The situation when \(3\times 3\) nilpotent matrices have circular numerical ranges is completely described by D. S. Keeler, L. Rodman and I. M. Spitkovsky [Linear Algebra Appl. 252, 115–139 (1997; Zbl 0876.15020)]. The \(4\times 4\) case is covered by H.-L. Gau [Taiwanese J. Math. 10, No. 1, 117–128 (2006; Zbl 1099.15022)] as follows: A necessary and sufficient condition that the numerical range of a nilpotent matrix \(M\) of size at most 4 be circular is that the traces \(\text{tr} M^*M^2\) and \(\text{tr} M^*M^3\) be all null.
For matrices of size 5, the situation becomes more complicated. The authors show that the analog of the so-called trace condition appearing in the result by Gau does not hold in the \(5\times 5\) case, counterexamples being given. The main result of the paper completely characterizes the situation when a \(5\times 5\) nilpotent matrix has circular numerical range. Also, the case when a \(5\times 5\) nilpotent matrix satisfying, respectively non-satisfying, the trace condition has circular numerical range is discussed thoroughly.
For matrices of size 5, the situation becomes more complicated. The authors show that the analog of the so-called trace condition appearing in the result by Gau does not hold in the \(5\times 5\) case, counterexamples being given. The main result of the paper completely characterizes the situation when a \(5\times 5\) nilpotent matrix has circular numerical range. Also, the case when a \(5\times 5\) nilpotent matrix satisfying, respectively non-satisfying, the trace condition has circular numerical range is discussed thoroughly.
Reviewer: Alexey Alimov (Moskva)
MSC:
| 15A60 | Norms of matrices, numerical range, applications of functional analysis to matrix theory |
References:
| [1] | Brown, E. S.; Spitkovsky, I. M., On flat portions on the boundary of the numerical range, Linear Algebra Appl., 390, 75-109 (2004) · Zbl 1059.15031 |
| [2] | Gau, H.-L., Elliptical numerical ranges of 4×4 matrices, Taiwanese J. Math., 10, 1, 117-128 (2006) · Zbl 1099.15022 |
| [3] | Gustafson, K. E.; Rao, D. K.M., Numerical Range (1997), Springer-Verlag: Springer-Verlag New York, Heidelberg, Berlin |
| [4] | Halmos, P. R., A Hilbert Space Problem Book (1980), Springer-Verlag: Springer-Verlag New York, Heidelberg, Berlin · Zbl 0202.12801 |
| [5] | Karaev, M. T., The numerical range of a nilpotent operator on a Hilbert space, Proc. Amer. Math. Soc., 132, 8, 2321-2326 (2004), electronic · Zbl 1069.47005 |
| [6] | Keeler, D. S.; Rodman, L.; Spitkovsky, I. M., The numerical range of 3×3 matrices, Linear Algebra Appl., 252, 115-139 (1997) · Zbl 0876.15020 |
| [7] | Kippenhahn, R., Über den Wertevorrat einer Matrix, Mathematische Nactrichten, 6, 193-203 (1951) · Zbl 0044.16201 |
| [9] | Schulte-Herbrüggen, T.; Dirr, G.; Helmke, U.; Glaser, S. G., The significance of the C-numerical range and the local C-numerical range in quantum control and quantum information, Linear and Multilinear Algebra, 56, 1 and 2, 3-26 (2008) · Zbl 1134.81018 |
| [10] | Tso, S.-H.; Wu, P. Y., Matricial ranges of quadratic operators, Rocky Mountain J. Math., 29, 3, 1139-1152 (1999) · Zbl 0957.47005 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
