Sublinear dimension growth in the Kreiss Matrix Theorem

We discuss a possible sublinear dimension growth in the Kreiss Matrix Theorem bounding the stability constant in terms of the Kreiss resolvent characteristic. Such a growth is proved for matrices having unimodular spectrum and acting on a uniformly convex Banach space. The principal ingredients to results obtained come from geometric properties of eigenvectors, where we use and compare the approaches by C. A. McCarthy–J. Schwartz (1965) and V. I. Gurarii–N. I. Gurarii (1971). The sharpness issue is verified via finite Muckenhoupt bases (by using mostly the approach by M. Spijker, S. Tracogna, and B. Welfert (2003)).