Complex strongly hyperbolic [formula omitted] systems, reduced dimension and hermitian systems

Abstract

a is a complex matrix valued 4 × 4 strongly hyperbolic operator; we state the following result: if the reduced real dimension of a is superior or equal to 13, a is hermitian in a convenient basis. We consider a matrix valued 4 × 4 differential operator, the coefficients of which are complex a ( D ) = I D 0 + a ( D ′ ) ; the real reduced dimension d ( a ) of the operator is the real dimension of the vector space of the matrices: { ξ 0 I + ∑ 1 ⩽ k ⩽ n ( Re a k + i Im a k ) ξ k ; ( ξ 0 , ξ ′ ) = ( ξ 0 , ξ 1 , … , ξ n ) ∈ R n + 1 } . We state the following theorem: if the reduced dimension: d ( a ) ⩾ 4 2 − 3 = 13 , if a ( D ) is strongly hyperbolic, then there exist a complex invertible matrix T, such that T −1 a ( ξ ) T is hermitian, ∀ ξ. The corresponding results for real coefficients and any m was studied in [J. Vaillant, Ann. Scuola Norm. Sup. Pisa Cl. Sc. 5 (1978) 405; Y. Oshime, J. Math. Kyoto Univ. 31 (1991) 937; T. Nishitani, Ann. Scuola Norm. Sup. Pisa Cl. Sc. 21 (1994) 97; J. Vaillant, Ann. Scuola Norm. Sup. Pisa Cl. Sc. 4 (XXIX) (2000) 839; J. Vaillant, Bull. Soc. Roy. Sc. Liege 70 (4–6) (2001) 407; J. Vaillant, in: Proc. 3rd International ISAAC Congress, World Scientific, 2003, pp. 1073–1080; J. Vaillant, in: Partial Diff. Equations and Math. Physics, in memory of Jean Leray, Birkhäuser, 2003, pp. 195, 223]. In the case of complex coefficients, if we replace the assumption of strong hyperbolicity by the assumption of real diagonalizability, if m ⩾ 3 , if d ( a ) ⩾ m 2 − 2 , the theorems were stated in [Y. Oshime, Publ. Res. Inst. Math. Sc. Kyoto Univ. 28 (1992) 223; J. Vaillant, in: Hyperbolic and Related Topics, International Press, 2003, pp. 403–422; J. Vaillant, Stud. Math. Bulgar. 15 (2003) 133]. In the case of real coefficients, the results were extended to the case of variable coefficients [T. Nishitani, J. Vaillant, Tsukuba J. Math. 25 (2001) 165; ibid. 27 (2003) 389]; the authors prepare the extension theorem to the case of complex variable coefficients.