Vector-valued versions of Fabry's ratio theorem (Q1921486)

Let \(A\) be a Banach algebra with an identity and \(r(a)\) be the spectral radius of \(a \in A\). The main result of the paper is the following Theorem 1. Let \((a_n)^\infty_{n = 0} \subset A\), \(a_i a_j = a_j a_i\), \(i,j \in \mathbb{N} \cup \{0\}\) and \(a_n \to a\), \(n \to \infty\). If \(r(a) = 0\) then the series \[ \sum^\infty_{n = 0} {a_0 a_1 \dots a_n \over \lambda^{n + 1}} \tag{1} \] converges for \(\lambda \neq 0\). If \(r(a) > 0\) and \(|a_n - a |< r(a)\) then the radius of convergence of the series (1) is equal to \(r(a)\) and \[ \lim_{n \to \infty} |a_0 a_1 \dots a_n |^{1/n} = r(a). \] For \(r(a) \neq 0\) each \(|\lambda |= r(a)\), where \(\lambda\) is the point of the spectrum of \(a\), is a singular point for the series (1).