In this paper, we study $(\alpha ,\beta )$-A-normal tuples of operators acting on semi-Hilbertian spaces, that is, Hilbert-like spaces endowed with a positive bounded operator A inducing a semi-inner product. By exploiting the geometric structure associated with A, we establish several operator inequalities and norm estimates that characterize this class of operator tuples. An A-characterization of $(\alpha ,\beta )$-A-normal tuples is obtained, and their stability properties are investigated. In particular, we show that this class is stable under the A-adjoint, invariant under similarity transformations induced by A-unitary operators, and stable under sums and products under suitable conditions. These results extend a number of classical inequalities from the Hilbert space setting to the semi-Hilbertian framework and contribute to the development of multivariable operator inequalities, with potential applications to joint spectral theory and numerical radius estimates.

Dilation matrices are important in multiple subdivision and multiple multiresolution analysis, as they govern the process of data refinement and play a crucial role in capturing directional features. One common limitation in the existing methods is the relatively large determinant of their dilation matrices, leading to high computational and storage costs. To address this issue, this paper proposes a novel family of pairs of directional dilation matrices with determinant 3. Such dilation matrices satisfy the joint expansion property and directional sensitivity. The joint expansion property is verified via the joint spectral radius, while by connecting the action of the matrices to certain elliptic elements of PSL(2,R), their directional adaptability can be established. Compared to most of the existing dilation matrices, the proposed ones achieve a balance between determinant and directional adaptability and provide a new insight into the construction of directional dilation matrices. This makes them suitable for addressing practical anisotropic problems.

In this paper we propose a new method to determine the joint spectral radius of a finite set of real matrices by verifying that a given family of candidates actually consists of spectrum maximizing products. Our algorithm aims at constructing a finite set-valued tree according to the approach of Möller and Reif using a norm that is constructed in the spirit of the invariant polytope algorithm. This combines the broad range of applicability of the first algorithm with the efficiency of the latter. • Novel approach for joint spectral radius computation. • Algorithm remains rigorous. • Algorithm has more general termination guarantees than classical invariant polytope algorithm.

The objective of this paper is to present a generalization of the concept of Euclidean joint spectral radius for tuples of Hilbert space operators, denoted as r p , h ( ⋅ ) , where p ∈ [ 1 , ∞ ] . One of the main goals of this study is to establish the equality between the well-known p-joint spectral radius, denoted as r p ( T ) , and r p , h ( T ) , where T = ( T 1 , … , T d ) represents a commuting operator tuple. This result enables us to establish various connections between the p-joint spectral radius, the p-joint numerical radius, and other p-joint norms documented in the literature. Additionally, we introduce new classes of multivariable operators and explore their applications.

We show that the joint spectral radius is pointwise Hölder continuous. In addition, the joint spectral radius is locally Hölder continuous for ε-inflations. In the two-dimensional case, local Hölder continuity holds on the matrix sets with positive joint spectral radius.

A bound on the joint spectral radius using the diagonals

We present an interpretation of switching signals with certain average dwell time in terms of Poisson process, which allows to build a probabilistic framework to accommodate all the switching signals of this kind. As a result, we can learn more knowledge about such signals. In particular, such a switching signal and the solution of the corresponding switching system jointly constitute a Markov process. And the Markov property will facilitate the asymptotic behaviour analysis of the switching dynamics. As a byproduct, we present an upper bound on the joint spectral radius of a family of Hurwitz or anti-Hurwitz matrices, which turns out to be quite sharp as compared with the existing results.

Learning stability of partially observed switched linear systems

Abstract This paper investigates the stability of threshold autoregressive models. We review recent research on stability issues from both a theoretical and empirical standpoint. We provide a sufficient condition for the stationarity and ergodicity of threshold autoregressive models by applying the concept of joint spectral radius to the switching system. The joint spectral radius criterion offers a generally applicable criterion to determine the stability in a threshold autoregressive model.

For a single matrix (operator) it is well-known that the spectral gap is an important quantity, as well as its estimate and computation. Here we consider, for the first time in the literature, the computation of its extension to a finite family of matrices, in other words the difference between the joint spectral radius (in short JSR, which we call here the first Lyapunov exponent) and the second Lyapunov exponent (denoted as SLE). The knowledge of joint spectral characteristics and of the spectral gap of a family of matrices is important in several applications, as in the analysis of the regularity of wavelets, multiplicative matrix semigroups and the convergence speed in consensus algorithms. As far as we know the methods we propose are the first able to compute this quantity to any given accuracy. For computation of the spectral gap one needs first to compute the JSR. A popular tool that is used to this purpose is the invariant polytope algorithm, which relies on the finiteness property of the family of matrices, when this holds true. In this paper we show that the SLE may not possess the finiteness property, although it can be efficiently approximated with an arbitrary precision. The corresponding algorithm and two effective estimates are presented. Moreover, we prove that the SLE possesses a weak finiteness property, whenever the leading eigenvalue of the dominant product is real. This allows us to find in certain situations the precise value of the SLE. Numerical results are demonstrated along with applications in the theory of multiplicative matrix semigroups and in the wavelets theory.

On maximal autocorrelations of Rudin–Shapiro sequences

Abstract We prove new mixing rate estimates for the random walks on homogeneous spaces determined by a probability distribution on a finite group$G$. We introduce the switched random walk determined by a finite set of probability distributions on$G$, prove that its long-term behaviour is determined by the Fourier joint spectral radius of the distributions, and give Hermitian sum-of-squares algorithms for the effective estimation of this quantity.

We prove new inequalities and equalities for the generalized and the joint spectral radius (and their essential versions) of Hadamard (Schur) geometric means of bounded sets of positive kernel operators on Banach function spaces. In the case of non-negative matrices that define operators on Banach sequences, we obtain additional results. Our results extend the results of several authors that appeared relatively recently.

On the joint spectral radius of nonnegative matrices

The spectral picture and joint spectral radius of the generalized spherical Aluthge transform

It is undecidable whether the growth rate of a given bilinear system is 1

Stability of shuffled switched linear systems: A joint spectral radius approach

In this paper we study ergodic optimization and multifractal behavior of Lyapunov exponents for matrix cocycles. We show that the restricted variational principle holds for generic cocycles [in the sense of (Bonatti and Viana in Ergod Theory Dyn Syst 24(5):1295–1330, 2004)] over mixing subshifts of finite type. We also show that the Lyapunov spectrum is equal to the closure of the set where the entropy spectrum is positive for such cocycles. Moreover, we show the continuity of the entropy spectrum at boundary of Lyapunov spectrum in the sense that h_{top}(E(alpha _{t})) rightarrow h_{top}(E(beta ({mathcal {A}})), where E(alpha )={xin X: lim _{nrightarrow infty }frac{1}{n}log Vert {mathcal {A}}^{n}(x)Vert =alpha }, for such cocycles. We prove the continuity of the lower joint spectral radius for linear cocycles under the assumption that linear cocycles satisfy a cone condition.

In this paper, we study the problem of estimating the state of a switched linear system (SLS), when the observation of the system is subject to communication constraints. We introduce the concept of worst-case topological entropy of such systems, and we show that this quantity is equal to the minimal data rate (number of bits per second) required for the state estimation of the system under arbitrary switching. Furthermore, we provide a closed-form expression for the worst-case topological entropy of switched linear systems, showing that its evaluation reduces to the computation of the joint spectral radius (JSR) of some lifted switched linear system obtained from the original one by using tools from multilinear algebra, and thus can benefit from well-established algorithms for the stability analysis of switched linear systems. Finally, drawing on this expression, we describe a practical coder-decoder that estimates the state of the system and operates at a data rate arbitrarily close to the worst-case topological entropy.

Abstract We consider the consensus of coupled harmonic oscillators with impulsive control via sampled position data in this article. Leader‐following consensus problem and leaderless consensus problem are investigated by unified analysis. Using joint spectral radius of the set of matrix, the consensus problem is solved without the assumption that the sampling interval is fixed. Some sufficient conditions which are valid for both cases of periodic and aperiodic sampled data control are obtained under a directed network topology. At last, some simulations are provided to support the theoretical analysis.