Joint similarity for commuting families of power bounded matrices

TL;DR

This paper proves that finite families of commuting power bounded matrices can always be jointly similar to contractions, extending the result to certain infinite families, contrasting with known infinite-dimensional counterexamples.

Contribution

It establishes that in finite dimensions, commuting power bounded matrices are always jointly similar to contractions, unlike the infinite-dimensional case.

Findings

Finite families of commuting power bounded matrices are jointly similar to contractions.

The result extends to certain infinite families with uniformity conditions.

Contrasts with Pisier's example in infinite dimensions.

Abstract

An example due to Pisier shows that two commuting, completely polynomially bounded Hilbert space operators may not be simultaneously similar to contractions. Thus, while each operator is individually similar to a contraction, the pair is not jointly similar to a pair of commuting contractions. We show that this phenomenon does not occur in finite dimensions. More precisely, we show that a finite family of power bounded commuting matrices is always jointly similar to a family of contractions. In fact, the result can be extended to infinite families satisfying certain uniformity conditions. Our approach is based on a joint spectral decomposition of the underlying space.