Polynomial numerical hulls of matrix polynomials

Abstract

Let M_n be the algebra of all n × n complex matrices. Suppose

is a matrix polynomial, where A_i  ∈ M_n (i=0,1, … ,m) and λ is a complex variable. In this note, we introduce the polynomial numerical hull and the joint polynomial numerical hull of order k of Q(λ). The emphasis is on the geometric properties of polynomial numerical hulls of matrix polynomials. Also, we study the relationships among spectra, numerical ranges and polynomial numerical hulls of order k of matrix polynomials.

§Dedicated to Professor Mehdi Radjabalipour for his 60th birthday.

Keywords:

• Polynomial numerical hull
• Matrix polynomial
• Joint polynomial numerical hull

Keywords:

• Subject Classification: 15A60
• 15A18

Acknowledgments

The authors would like to thank the referee for some useful comments. The research was supported by INSF grant #84041 and the second author is supported by the Azad university of Baft.

Notes

§Dedicated to Professor Mehdi Radjabalipour for his 60th birthday.