Higher rank numerical ranges of rectangular matrix polynomials

Subject Areas : Linear and multilinear algebra; matrix theory

Gh. Aghamollaei ¹ , M. Zahraei ²

1 - Department of Mathematics, Shahid Bahonar University of Kerman, 76169-14111, Kerman, Iran
2 - Department of Mathematics, Ahvaz Branch, Islamic Azad University, Ahvaz, Iran

Received: 2015-10-12 Accepted : 2015-12-28 Published : 2014-12-30

Keywords: isometry, numerical range, Rank-k numerical range, rectangular matrix polynomials,

Abstract :

In this paper, the notion of rank-k numerical range of rectangular complex matrix polynomials are introduced. Some algebraic and geometrical properties are investigated.Moreover, for ϵ > 0; the notion of Birkhoff-James approximate orthogonality sets for ϵ-higherrank numerical ranges of rectangular matrix polynomials is also introduced and studied. The proposed denitions yield a natural generalization of the standard higher rank numericalranges.

References:

[1] A. Aretaki and J. Maroulas, On the rank􀀀k numerical range of matrix polynomials, Electronic J. Linear Algebra, 27 (2014), 809-820.

 

[2] F.F. Bonsall and J. Duncan, Numerical Ranges of Operators on Normed Spaces and Elements of Normed Algebras, Cambridge University Press, New York, 1971.

 

[3] M.D. Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl. 10 (1975), 285-290.

 

[4] M.D. Choi, M. Giesinger, J.A. Holbrook and D.W. Kribs, Geometry of higher rank numerical ranges, Linear Multilinear Algebra, 56 (2008), 53-64.

 

[5] C. Chorianopoulos, S. Karanasios and P. Psarrakos, A denition of numerical range of rectangular matrices, Linear Multilinear Algebra, 51 (2009), 459-475.

 

[6] C. Chorianopoulos and P. Psarrakos, Birkhoff-James approximate orthogonality sets and numerical ranges, Linear Algebra Appl. 434 (2011), 2089-2108.

 

[7] S. Clark, C.K. Li and N.S. Sze, Multiplicative maps preserving the higher rank numerical ranges and radii, Linear Algebra Appl. 432 (2010), 2729-2738.

 

[8] I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, Academic Press, New York, 1982. 

 

[9] K.E. Gustafson and D.K.M. Rao, Numerical Range: The Field of Values of Linear Operators and Matrices, Springer-Verlage, New York, 1997.

 

[10] R. Horn and C. Johnson, Topics in Matrix Analysis, Cambridg University Press, New York, 1991.

 

[11] D.W . Kribs, R. Laflamme, D. Poulin and M. Lesosky, Operator quantum error correction, Quant. Inf. Comput. 6 (2006), 383-399.

 

[12] P. Lancaster and M. Tismenetsky, The Theory of Matrices, Academic Press, Orland, 1985.

 

[13] C.K. Li and L. Rodman, Numerical range of matrix polynomials, SIAM J. Matrix Anal. Appl. 15 (1994), 1256-1265.

 

[14] J.G. Stampi and J.P. Williams, Growth conditions and the numerical range in a Banach algebra, T. Math. J. 20 (1968), 417-424.

 

[15] M. Zahraei and Gh. Aghamollaei, Higher rank numerical ranges of rectangular matrices, Ann. Func. Anal. 6(2015), 133-142.