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A new inequality about matrix products and a Berger-Wang formula. (Une nouvelle inégalité sur LES produits de matrices et une formule de Berger-Wang.) (English. French summary) Zbl 1437.15030

The Beruling-Gelfand spectral radius formula asserts that for each \(A\in M_d(\mathbb{C})\), \[ \lim_{m\to \infty} \|A^m\|^{1/m} = \rho(A),\] where \(\rho(A)\) is the spectral radius of \(A\) and \(\|\cdot\|\) is the spectral norm. The formula is also valid for any matrix norm, i.e., a norm with submultiplicative property \(\||AB\||\le \||A\|\||B\||\) for all \(A, B \in M_d(\mathbb{C})\),
Let \(\|\cdot\|\) be the operator norm \(M_d(\mathbb{C})\) induced by a vector norm \(\|\cdot\|\) on \(\mathbb{C}^n\). The Berger-Wang formula (see [M. A. Berger and Y. Wang, Linear Algebra Appl. 166, 21–27 (1992; Zbl 0818.15006)]) is an extension of Beruling-Gelfand formula and asserts that if \({\mathcal M}\subset M_d(\mathbb{C})\) is a bounded set, then \[\mathrm R({\mathcal M}) = {\lim\sup}_{n\to \infty} (\sup \{\rho(A_1\cdots A_n): A_i\in {\mathcal M}\})^{1/n}\] where \[\mathrm R({\mathcal M}):= \lim_{n\to \infty} (\sup\{\|A_1 \cdots A_n\|: A_i\in {\mathcal M}\})^{1/n}\] is called the joint spectral radius of \({\mathcal M}\), introduced by G.-C. Rota and W. G. Strang [Nederl. Akad. Wet., Proc., Ser. A 63, 379–381 (1960; Zbl 0095.09701)]. It represents the maximal exponential growth rate of the partial sequence of products \((A_1\cdots A_n)_n\) of a sequence of matrices \(A_1, A_2, \dots, \) with \(A_i\in \mathcal M\). The quantity \(\sup \{\rho(A_1\cdots A_n): A_i\in {\mathcal M}\}\) is called the generalized spectral radius. Along with several results, the author proves the following:
Theorem. Let \(d\in \mathbb{N}\), \(k\) be a local field, and \(\|\cdot\|\) be matrix norm on \(M_d(k)\). There exist constants \(N=N(d)\le \prod_{i=1}^d \binom di\), \(r=r(d,N)\le (Nd+1)^{Nd^2+2}\) and \(C =C(d,\|\cdot\|)>1\) such that for all \(n\ge N\) and \(A_1, \dots, A_n\in M_d(k)\): \[ \|A_n\cdots A_1\|\le C \left(\prod_{1\le i\le n} \|A_i\| \right) \max_{1\le \alpha \le \beta \le n} \left( \frac {\rho(A_\beta\cdots A_\alpha)}{\prod_{\alpha\le i\le \beta} \|A_i\|}\right)^{1/r},\tag{1} \] where the right side is treated as zero if one of the \(A_i\) is zero.
Thus, for large \(n\), if \(\|A_n\cdots A_1\|\) is comparable to \(\prod_{1\le i\le n} \|A_i\|\), then there exist s sub-product \(A_\beta, \dots, A_\alpha\) whose spectral radius is comparable to \(\prod_{\alpha\le i\le \beta} \|A_i\|\).
The Beruling-Gelfand formula is used in the proof of Theorem 1.2 to get a uniform bound for the multiplicative constant in the general case of matrix norms, and is not needed for the case of operator norms. The proof of the inequality is based on the non trivial case of equality, where the right hand side is zero but the matrices \(A_i\) are nonzero. This particular case can be stated in the context of nilpotent matrices in Theorem 1.3 which is related to some nice theorems such as Levitzki’s theorem or Burnside-Schur theorem. An interesting polynomial identity is obtained in the process.
Remark: The author proves (see Proposition 4.2) that when \(k=\mathbb{C}\), there is \(C =C(d)>1\) such that (1) holds for all operator norms \(\|\cdot\|\) on \(M_d(\mathbb{C})\) and \(A_1, \dots, A_n\in M_d(\mathbb{C})\). The reviewer would like to point out that it is true for all matrix norms as Lemma 4.3 is also true for matrix norms \(\||\cdot\||\) because \(\||SAS^{-1}\||\) (see Lemma 4.3) is also a matrix norm for each nonsingular \(S\in M_d(\mathbb{C})\). It is worth to observe that operator norms are induced by vector norms and are matrix norms (see, e.g.,Chapter 5 of [R. A. Horn and C. R. Johnson, Matrix analysis. 2nd ed. Cambridge: Cambridge University Press (2013; Zbl 1267.15001)]).
Reviewer: Tin Yau Tam (Reno)

MSC:

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A45 Miscellaneous inequalities involving matrices
37H15 Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
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