arXiv subjects
Publications and source records attributed to A. Peperko.
We prove new inequalities for the essential generalized and the essential joint spectral radius of Hadamard (Schur) weighted geometric means of bounded sets of infinite nonnegative matrices that define operators on suitable Banach sequence spaces and of bounded sets of positive kernel operators on $L^2$. To our knowledge the obtained inequalities are new even in the case of singelton sets.
In this article we introduce a new method, which we call a mutation-sunflower method, for calculating max-eigenvectors of a nonnegative irreducible $n\times n$ matrix $A$. Our method works in the general irreducible case, but it is in comparison with existing methods most effective for some special classes of matrices for example for sparse enough matrices. Our method reduces to solving max-eigenproblems for simple mutation-sunflower matrices that have exactly one positive entry in each row. We include some instructive examples.
We prove new explicit asymptotic formulae between (geometric) eigenvalues in max-algebra and classical distinguished eigenvalues of nonnegative matrices, which are useful tools for transferring results between both settings. We establish new inequalities for both types of eigenvalues of Hadamard products and Hadamard weighted geometric means of nonnegative matrices. Moreover, a version of the spectral mapping theorem for the distinguished spectrum is pointed out.
Let $\Psi$ and $\Sigma$ be bounded sets of positive kernel operators on a Banach function space $L$. We prove several refinements of the known inequalities $$\rho \left(\Psi ^{\left( \frac{1}{2} \right)} \circ \Sigma ^{\left( \frac{1}{2} \right)} \right) \le \rho (\Psi \Sigma) ^{\frac{1}{2}} \;\; \mathrm{and}\;\; \hat{\rho} \left(\Psi ^{\left( \frac{1}{2} \right)} \circ \Sigma ^{\left( \frac{1}{2} \right)} \right) \le \hat{\rho} (\Psi \Sigma) ^{\frac{1}{2}} $$ for the generalized spectral radius $\rho$ and the joint spectral radius $\hat{\rho}$, where $\Psi ^{\left( \frac{1}{2} \right)} \circ \Sigma ^{\left( \frac{1}{2} \right)} $ denotes the Hadamard (Schur) geometric mean of the sets $\Psi $ and $\Sigma$. Furthermore, we prove that analogous inequalities hold also for the generalized essential spectral radius and the joint essential spectral radius in the case when $L$ and its Banach dual $L^*$ have order continuous norms.