On the limit products of a family of matrices (Q1863564)

The authors [Linear Algebra Appl. 322, 169-192 (2001; Zbl 0971.15016)] introduced the concept of the limit spectrum-maximizing product (l.s.m.p.) for a bounded family of complex \(n\times n\) matrices and characterized the defectivity of finite families of \(2\times 2\) matrices by the existence of defective l.s.m.\ products. At the end of [loc. cit.] it was conjectured that this characterization should hold in general. Here the authors prove that all nondefective bounded normalized families of matrices always have an l.s.m.p.\ and show that finite defective families of \(3\times 3\) matrices always have an spectrum-maximizing product (s.m.p.), but may not have a defective l.s.m.p. Thus, the conjecture occured to be false. Also, the authors construct a finite defective family of \(4\times 4\) matrices which does not have even any l.s.m.p., regardless its possible defectivity.