Dedicated to the memory of Lior Tzafriri MINIMALITY PROPERTIES OF TSIRELSON TYPE SPACES

In this paper, we study minimality properties of partly modified mixed Tsirelson spaces. A Banach space with a normalized basis (ek) is said to be subsequentially minimal if for every normalized block basis (xk) of (ek), there is a further block basis (yk) of (xk) such that (yk) is equivalent to a subsequence of (ek). Sufficient conditions are given for a partly modified mixed Tsirelson space to be subsequentially minimal and connections with Bourgain’s `-index are established. It is also shown that a large class of mixed Tsirelson spaces fails to be subsequentially minimal in a strong sense. The class of mixed Tsirelson spaces plays an important role in the structure theory of Banach spaces and has been well investigated (e.g., [2, 3, 5, 17, 20, 21]). In this paper, we will study aspects of the subspace structure of mixed Tsirelson spaces and (partly) modified mixed Tsirelson spaces (see definitions below). We are particularly interested in properties connected with minimality. A infinite-dimensional Banach space X is minimal if every infinite-dimensional subspace has a further subspace isomorphic to X. The work of Gowers [15] had motivated some recent studies on minimality (e.g., [11], [12], [22]). A Banach space X with a normalized basis (ek) is said to be subsequentially minimal if for every normalized block basis (xk) of (ek) , there is a further block (yk) of (xk) such that (yk) is equivalent to a subsequence of (ek) . It is well known that the Tsirelson space T [(S1, 1/2)] has the property that every normalized block basis of its standard basis is equivalent to a subsequence of (ek) [8]. In particular, it is subsequentially minimal. In [18, Theorem 9], it was shown that if a nonincreasing null sequence (θn) in (0, 1) is regular (θm+n ≥ θmθn) and satisfies (†) lim m lim sup n θm+n θn > 0, 2000 Mathematics Subject Classification. 46B20; 46B45.