More mixed Tsirelson spaces that are not isomorphic to their modified versions

Abstract

The class of mixed Tsirelson spaces is an important source of examples in the recent development of the structure theory of Banach spaces. The related class of modified mixed Tsirelson spaces has also been well studied. In the present paper, we investigate the problem of comparing isomorphically the mixed Tsirelson space $T[({\mathcal{S}}_{n},\theta_{n})_{n=1}^{\infty}]$ and its modified version $T_{M}[({\mathcal{S}}_{n},\theta_{n})_{n=1}^{\infty }]$. It is shown that these spaces are not isomorphic for a large class of parameters $(θ_n)$.