Disjoint supercyclicity in Banach algebras
Stefan Ivkovic
TL;DR
The paper develops a general framework to characterize disjoint F-semi-transitivity and disjoint transitivity for operators on non-unital Banach algebras that act as left multipliers composed with isometric automorphisms. It proves a central equivalence: such operator families are dF-semi-transitive precisely when a constructive finite-approximation criterion involving p_alpha and alpha-inverses can be satisfied with controlled cross-terms. The authors then instantiate the theory in several concrete settings—generalized weighted shifts on Hilbert C*-modules, operator-valued weighted composition operators on spaces of continuous functions, and weighted shifts on Segal-type function spaces—providing explicit criteria and illustrative examples. This unifies and extends disjoint transitivity phenomena across diverse operator-algebra contexts and yields practical conditions for verifying disjoint hypercyclicity-type properties. The work broadens the understanding of dynamic properties of operator families in non-unital Banach algebras and offers a versatile toolkit for constructing and analyzing disjoint transitivity in functional-analytic settings.
Abstract
In this paper, we consider operators that are a composition of an isometric isomorphism and a left multiplier on a normed algebra, and we characterize disjoint F-semi-transitive and disjoint supercyclic such operators on a large class of non-unital normed algebras. It turns out that generalized weighted bilateral shifts on the standard Hilbert C*-module are just a special case of our theory. Generalized weighted composition operators on the normed algebra of operator-valued continuous functions vanishing at infinity on a locally compact, non-compact Hausdorff space are another special case of our theory. Also, we illustrate our results with concrete examples.