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Continuity properties of the lower spectral radius. (English) Zbl 1311.15022

Let \(\mathcal{K}:=\mathcal{K}(\mathrm{GL}_{d}(\mathbb{R))}\) be the set of all compact, nonempty sets of \(d\times d\) invertible real matrices. The Hausdorff metric on \(\mathcal{K}\) is defined by \[ d_{H}(\mathbf{A},\mathbf{B}):=\max\left\{ \sup_{A\in\mathbf{A}}\inf_{B\in\mathbf{B}}\left\| A-B\right\| ,\sup_{B\in\mathbf{B}}\inf_{A\in\mathbf{A}}\left\| A-B\right\| \right\}, \] where \(\left\| \cdot \right\| \) is the Euclidean operator norm. The lower spectral radius of a set \(\mathbf{A\in}\mathcal{K}\) is defined to be \[ \check{\rho}(\mathbf{A}):=\lim_{n\rightarrow\infty}\inf\left\{ \left\| A_{n} \dots A_{1}\right\| ^{1/n} \mid \text{each }A_{i}\in\mathbf{A}\right\} \] (see [L. Gurvits, Linear Algebra Appl. 231, 47–85 (1995; Zbl 0845.68067)]). This spectral radius arises naturally in contexts such as stability of linear inclusions in control theory, regularity of fractal structures and combinatorics on words. There is an analogous function \(\hat{\rho}\) (where inf is replaced by sup) called the upper spectral radius which is known to be Lipschitz continuous at \(\mathbf{A}\) with respect to \(d_{H}\) whenever \(\mathbf{A}\) admits no nontrivial invariant subspace [F. Wirth, Linear Algebra Appl. 342, No. 1–3, 17–40 (2002; Zbl 0996.15020)]. The object of the present paper is to investigate continuity properties of \(\check{\rho}\).
For any \(A\in \mathrm{GL}_{d}(\mathbb{R})\) let \(\sigma_{1}(A)\geq\sigma_{2} (A)\geq \dots \geq\sigma_{d}(A)\) denote the singular values of \(A\). Suppose that \(\mathbf{A}\in\mathcal{K}\) and \(1\leq k<d\). Then, \(\mathbf{A}\) is \(k\)-dominated if there exist constants \(C>1\) and \(\tau\) with \(0<\tau<1\) such that for every sequence \(\left\{ A_{i}\right\} \) in \(\mathbf{A}\) we have \(\sigma _{k+1}(A_{n} \dots A_{1})\leq C\tau^{n}\sigma_{k}(A_{n} \dots A_{1})\) for every \(n\geq1\). Let \(\ell(\mathbf{A})\) be the least \(k\) such that \(\mathbf{A}\) is \(k\)-dominated (\(\ell(\mathbf{A})=d\) if there is no smaller \(k\)). The following are some typical results. (Theorem 1.5) Let \(\mathcal{D}\) be the set of all \(1\)-dominated \(\mathbf{A}\in\mathcal{K}\). Then, \(\check{\rho}\) is locally Lipschitz continuous on \(\mathcal{D}\). (Corollaries 1.7 and 1.8): \(\check {\rho}:\mathcal{K}\rightarrow\mathbb{R}\) is continuous at \(\mathbf{A}\) if and only if \(\check{\rho}(A)=\check{\rho}(\wedge^{\ell(\mathbf{A})}\mathbf{A} )^{1/\ell(\mathbf{A})}\), where \(\wedge^{\ell(\mathbf{A})}\) denotes the \(\ell(\mathbf{A})\) exterior power; in particular\(, \check{\rho}\) is continuous at any singleton. There are many other results and the paper concludes with a series of open questions.

MSC:

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
47D03 Groups and semigroups of linear operators
15A18 Eigenvalues, singular values, and eigenvectors
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