A Pringsheim-type convergence criterion for continued fractions in Banach algebras

We consider continued fractions in Banach algebras, that is b0+a1(b1+a2(b2+⋯)−1)−1, where (bn)n∈N0 and (an)n∈N are sequences of elements of some Banach algebra. We prove that ‖bn−1‖+‖anbn−1‖≤1 for n=1,2,… is a sufficient condition for convergence. This result is an exact generalization of the Śleszyński-Pringsheim convergence criterion for complex continued fractions, and improves on all known results.