A Perron-Frobenius type result in Banach algebras via asymptotic closeness to a cone (Q6594787)

The authors present a sufficient condition for the inclusion of the spectral radius of an element of a Banach algebra in its spectrum. Let \(\mathcal A\) be a complex Banach algebra with unit \( 1\). The symbols \(r(a)\) and \(\sigma(a)\) denote the spectral radius and, respectively, the spectrum of \(a\in{\mathcal A}\). The algebra \(\mathcal A\) is equipped with a standard partial order \(\leq\) generated by a cone \(K\) (i.e., \(K\) is a nonempty and closed subset of \(\mathcal A\) with \(K+K\subset K\), \(\lambda K\subset K\) for \(\lambda \geq 0\) and \(K\cap (-K)=\{0\}\), and \(a\leq b\) if and only if \(b-a\in K\)). Moreover it is assumed that the cone \(K\) is normal, i.e., there exists \(\gamma\geq 1\) such that for all \(a,b\in{\mathcal A}\) with \(0\leq a\leq b\) it is \(\|a\|\leq \gamma \|b\|\). The main result of the paper reads as follows: \N\NTheorem. If \(a\in{\mathcal A}\) with \(r(a)>0\) satisfies the condition \N\[\N \limsup_{k\to\infty}\left(\mathrm{dist}\left(\dfrac{a^k}{r(a)^k},K\right)\right)<\dfrac{1}{\pi\gamma+1}, \tag{\({\ast}\)}\N\]\Nthen \(r(a)\in\sigma(A)\).\N\NInequality \((\ast)\) is a sort of a Perron-Frobenius condition. Several variants of this condition have appeared in the theories of ordered Banach algebras, positive matrices, and positive linear operators.