On matrix semigroups bounded above and below

TL;DR

This paper characterizes when an irreducible norm-closed semigroup of complex matrices can be transformed into a semigroup of partial isometries, based on boundedness and commutativity of idempotents.

Contribution

It generalizes the classical result on bounded groups to a broader class of matrix semigroups, identifying key conditions for similarity to partial isometries.

Findings

Semigroup is similar to partial isometries if norms are bounded and idempotents commute.

Provides necessary and sufficient conditions for such similarity.

Extends known results from groups to more general semigroups.

Abstract

An irreducible norm closed semigroup of complex matrices is simultaneously similar to a semigroup of partial isometries if and only if (a) the norms of all nonzero members of it are uniformly bounded above and below, and (b) its idempotents commute. This is a generalization of the well-known result on bounded groups.