In this paper we direct attention at bounded families of complex n×n-matrices. In order to study their asymptotic behaviour, we recall from [Linear Algebra Appl. 322 (2001) 162] the concept of limit spectrum-maximizing product and show that nondefective families always admit such limit products. Then we consider defective families. In [loc. cite] we proved that, for finite families of 2×2-matrices, defectivity is equivalent to the existence of defective such limit products. This result led us to conjecture the validity of this property also for higher dimensions n>2. Here, instead, by making use of the results obtained by Bousch and Mairesse [J. Am. Math. Soc. 15 (2002) 77] that disproved the well-known Finiteness Conjecture, we find some counterexamples to our conjecture in [loc. cite] for all n>2.
On the limit products of a family of matrices
ZENNARO, MARINO
2003-01-01
Abstract
In this paper we direct attention at bounded families of complex n×n-matrices. In order to study their asymptotic behaviour, we recall from [Linear Algebra Appl. 322 (2001) 162] the concept of limit spectrum-maximizing product and show that nondefective families always admit such limit products. Then we consider defective families. In [loc. cite] we proved that, for finite families of 2×2-matrices, defectivity is equivalent to the existence of defective such limit products. This result led us to conjecture the validity of this property also for higher dimensions n>2. Here, instead, by making use of the results obtained by Bousch and Mairesse [J. Am. Math. Soc. 15 (2002) 77] that disproved the well-known Finiteness Conjecture, we find some counterexamples to our conjecture in [loc. cite] for all n>2.Pubblicazioni consigliate
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