Roots in operator and Banach algebras

TL;DR

This paper extends classical matrix root results to operator and Banach algebras, demonstrating convergence of iterative methods and generalizing concepts like sign and geometric mean.

Contribution

It introduces new generalizations of matrix root properties and iterative methods to the broader context of operator and Banach algebras.

Findings

Newton, binomial, Visser, and Halley methods converge in Banach and operator algebras.

Sign and geometric mean concepts are extended and analyzed in these algebras.

Several properties of roots are established in the operator and Banach algebra setting.

Abstract

We show that several known facts concerning roots of matrices generalize to operator algebras and Banach algebras. We show for example that the so-called Newton, binomial, Visser, and Halley iterative methods converge to the root in Banach and operator algebras under various mild hypotheses. We also show that the `sign' and `geometric mean' of matrices generalize to Banach and operator algebras, and we investigate their properties. We also establish some other facts about roots in this setting.