Some results on the block numerical range

Document Type: Research Paper


1 University of Hormozgan

2 Vali-e-Asr University of Rafsanjan



The main results of this paper are generalizations of classical results from the numerical range to the block numerical range.
A different and simpler proof for the Perron-Frobenius theory on the block numerical range of an irreducible nonnegative matrix is given.
In addition, the Wielandt's lemma and the Ky Fan's theorem on the block numerical range are extended.


[1] K.H. Forster and  N. Hartanto, On the block numerical range of nonnegative matrices, Oper. Theory: Adv. Appl., 188 (2008),  113-133.
[2] K.E. Gustafson and D.K.M. Rao, Numerical range: The field of values of linear operators and matrices, Springer-Verlag,  New York,  1997.
[3] R.A. Horn and  C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.
[4] R.A. Horn and C.R. Johnson, Topics in Matrix Analysis,  Cambridge University Press, Cambridge, 1991.
[5] J.N. Issos, The Field of Values of Non-Negative Irreducible Matrices, Ph.D. Thesis, Auburn University, 1966.
[6]H. Langer and C. Tretter, Spectral decomposition of some nonselfadjoint block operator matrices, J. Oper. Theory, 39(2) (1998),  339-359.
[7] C.K. Li, B.S. Tam and P.Y. Wu, The numerical range of a nonnegative matrix, Linear Algebra Appl., 350 (2002), 1-23.
[8] J. Maroulas, P.J. Psarrakos and M.J. Tsatsomeros, Perron-Frobenius type results on the numerical range, Linear Algebra Appl., 348 (2002), 49--62.
[9] H. Mink, Nonnegative Matrices, Wiley, New York, 1988.
[10] A. Salemi, Total decomposition and the block numerical range, Banach J. Math. Anal., 5(1) (2011), 51-55.
[11] C. Tretter and M. Wagenhofer, The block numerical range of an $ ntimes n $ block operator matrix, SIAM J. Matrix Anal. Appl., 24(4) (2003),  1003--1017.