##
**An infinite Ramsey theorem and some Banach-space dichotomies.**
*(English)*
Zbl 1030.46005

One of the main results of this deep paper (which circulated several years among specialists in preprint form) is the following dichotomy (Theorem 1.4): Every Banach space contains a subspace which either has an unconditional basis or is hereditarily indecomposable. This result is applied to solve an old problem of S. Banach who asked if the separable Hilbert space is the only homogeneous Banach space (that is an infinite-dimensional Banach space isomorphic to any of its infinite-dimensional subspaces). Combining the above dichotomy with the results of W. Gowers and B. Maurey [J. Am. Math. Soc. 6, 851-874 (1993; Zbl 0827.46008)] (asserting that a hereditarily indecomposable Banach space is isomorphic to no proper subspace of itself) and of R. Komorowski and N. Tomczak-Jaegermann [Isr. J. Math. 89, 205-226 (1995; Zbl 0830.46008)] (asserting that each homogeneous Banach space either is isomorphic to a Hilbert space or else fails to have an uncounditional basis) the author concludes that each homogeneous Banach space is isomorphic to the separable Hilbert space, thus answering the Banach problem affirmatively.

The mentioned dichotomy is a consequence of another Banach space dichotomy that has Ramsey-theoretic nature and by its spirit is close to the classical Nash-Williams Theorem [Proc. Camb. Philos. Soc. 61, 33-39 (1965; Zbl 0129.00602)]. At first some notation. Given a Banach space \(X\) with a fixed basis let \(\Sigma_f\) be the set of all finite block bases, that is sequences \(x_1<\dots<x_n\) of non-zero vectors from the unit ball of \(X\), where \(x<y\) means that \(\max\text{supp}(x)<\min\text{supp}(y)\). A subspace \(Y\) of \(X\) generated by an infinite block basis \(x_1<x_2<\dots\) is called a block subspace of \(X\). The second Banach space dichotomy (Theorem 2.1 proved in the second and third section) asserts that given a Banach space \(X\) with a fixed basis, a family \(\sigma\subset\Sigma_f\), and an infinite sequence \(\Delta=(\delta_n)\) of positive reals, one can find a block subspace \(Y\) of \(X\) such that either no sequence \(y_1<\dots<y_n\) of vectors of \(Y\) belongs to \(\sigma\) or else the second player has a winning strategy in the following infinite two-person game \(\sigma_\Delta[Y]\): At the \(n\)-th inning the first player selects an infinite-dimensional block subspace \(Y_n\) of \(Y\) while the second player responds with a vector \(y_n>y_{n-1}\) in \(Y_n\). The second player wins the game \(\sigma_\Delta[Y]\) if for some \(n\) there is a sequence \(x_1<\dots<x_n\) in \(\sigma\) such that \(\|y_i-x_i\|\leq\delta_i\) for all \(i\leq n\).

The main result of the fourth and fifth section is Theorem 4.1, which is an infinite version of Theorem 2.1 (the principal difference is that \(\Sigma_f\) is replaced by a collection \(\Sigma\) of block bases, and \(\sigma\) is an analytic subspace of \(\Sigma\) with respect to a suitable Polish topology on \(\Sigma\)). In section 6 for a Banach space \(X\supset c_0\), Theorem 4.1 is strengthened as follows (Theorem 6.5): For any sequence \(\Delta=(\delta_n)\) of positive reals and any analytic subset \(\sigma\) of the space \(\Sigma_1(X)\) of normalized block bases (equipped with a suitable Polish topology) there is a block subspace \(Y\) of \(X\) such that either \(\sigma\cap\Sigma_1(Y)=\emptyset\) or else for each block basis \((y_n)\) in \(Y\) there is a block basis \((x_n)\) in \(\sigma\) with \(\|y_i-x_i\|\leq\delta_n\) for all \(n\in\mathbb N\). In the Appendix the author proves that Theorem 6.5 fails for Banach spaces containing no isomorphic copy of \(c_0\) and also shows that the analyticity of \(\sigma\) is essential in Theorem 6.5 as well as in Theorem 4.1.

The seventh section is devoted to applications of the Ramsey Theorem 4.1 and contains the following dichotomy (Theorem 7.2): Each Banach space \(X\) with unconditional basis contains a block subspace \(Y\) such that either any two subspaces of \(Y\) have isomorphic infinite-dimensional subspaces or else any two disjointly supported subspaces of \(Y\) fail to be isomorphic. In the latter case \(Y\) is isomorphic to no proper subspace of itself. Moreover, for each isomorphism \(T:Z\to W\) between block subspaces of \(Y\) there is an invertible diagonal operator \(D:Y\to Y\) such that the operators \(T-D|_Z\) and \(T^{-1}-D^{-1}|_Z\) are strictly singular. It is interesting to compare this result with a result of V. Ferenczi [Bull. London Math. Soc 29, 338-344 (1997; Zbl 0899.46010)] asserting that each operator from a subspace of a hereditarily indecomposable Banach space is a strictly singular perturbation of a multiple of the inclusion map.

In the final section “Recent Development” the author makes some interesting historical notes and also refers the reader to the papers by J. Bagaria and J. López-Abad [Adv. Math. 160, 133-174 (2001; Zbl 0987.46014) and Trans. Am. Math. Soc. 354, 1327-1349 (2002; Zbl 0987.03040)] containing some related results and generalizations.

The mentioned dichotomy is a consequence of another Banach space dichotomy that has Ramsey-theoretic nature and by its spirit is close to the classical Nash-Williams Theorem [Proc. Camb. Philos. Soc. 61, 33-39 (1965; Zbl 0129.00602)]. At first some notation. Given a Banach space \(X\) with a fixed basis let \(\Sigma_f\) be the set of all finite block bases, that is sequences \(x_1<\dots<x_n\) of non-zero vectors from the unit ball of \(X\), where \(x<y\) means that \(\max\text{supp}(x)<\min\text{supp}(y)\). A subspace \(Y\) of \(X\) generated by an infinite block basis \(x_1<x_2<\dots\) is called a block subspace of \(X\). The second Banach space dichotomy (Theorem 2.1 proved in the second and third section) asserts that given a Banach space \(X\) with a fixed basis, a family \(\sigma\subset\Sigma_f\), and an infinite sequence \(\Delta=(\delta_n)\) of positive reals, one can find a block subspace \(Y\) of \(X\) such that either no sequence \(y_1<\dots<y_n\) of vectors of \(Y\) belongs to \(\sigma\) or else the second player has a winning strategy in the following infinite two-person game \(\sigma_\Delta[Y]\): At the \(n\)-th inning the first player selects an infinite-dimensional block subspace \(Y_n\) of \(Y\) while the second player responds with a vector \(y_n>y_{n-1}\) in \(Y_n\). The second player wins the game \(\sigma_\Delta[Y]\) if for some \(n\) there is a sequence \(x_1<\dots<x_n\) in \(\sigma\) such that \(\|y_i-x_i\|\leq\delta_i\) for all \(i\leq n\).

The main result of the fourth and fifth section is Theorem 4.1, which is an infinite version of Theorem 2.1 (the principal difference is that \(\Sigma_f\) is replaced by a collection \(\Sigma\) of block bases, and \(\sigma\) is an analytic subspace of \(\Sigma\) with respect to a suitable Polish topology on \(\Sigma\)). In section 6 for a Banach space \(X\supset c_0\), Theorem 4.1 is strengthened as follows (Theorem 6.5): For any sequence \(\Delta=(\delta_n)\) of positive reals and any analytic subset \(\sigma\) of the space \(\Sigma_1(X)\) of normalized block bases (equipped with a suitable Polish topology) there is a block subspace \(Y\) of \(X\) such that either \(\sigma\cap\Sigma_1(Y)=\emptyset\) or else for each block basis \((y_n)\) in \(Y\) there is a block basis \((x_n)\) in \(\sigma\) with \(\|y_i-x_i\|\leq\delta_n\) for all \(n\in\mathbb N\). In the Appendix the author proves that Theorem 6.5 fails for Banach spaces containing no isomorphic copy of \(c_0\) and also shows that the analyticity of \(\sigma\) is essential in Theorem 6.5 as well as in Theorem 4.1.

The seventh section is devoted to applications of the Ramsey Theorem 4.1 and contains the following dichotomy (Theorem 7.2): Each Banach space \(X\) with unconditional basis contains a block subspace \(Y\) such that either any two subspaces of \(Y\) have isomorphic infinite-dimensional subspaces or else any two disjointly supported subspaces of \(Y\) fail to be isomorphic. In the latter case \(Y\) is isomorphic to no proper subspace of itself. Moreover, for each isomorphism \(T:Z\to W\) between block subspaces of \(Y\) there is an invertible diagonal operator \(D:Y\to Y\) such that the operators \(T-D|_Z\) and \(T^{-1}-D^{-1}|_Z\) are strictly singular. It is interesting to compare this result with a result of V. Ferenczi [Bull. London Math. Soc 29, 338-344 (1997; Zbl 0899.46010)] asserting that each operator from a subspace of a hereditarily indecomposable Banach space is a strictly singular perturbation of a multiple of the inclusion map.

In the final section “Recent Development” the author makes some interesting historical notes and also refers the reader to the papers by J. Bagaria and J. López-Abad [Adv. Math. 160, 133-174 (2001; Zbl 0987.46014) and Trans. Am. Math. Soc. 354, 1327-1349 (2002; Zbl 0987.03040)] containing some related results and generalizations.

Reviewer: Taras Banakh (Lviv)

### MSC:

46B03 | Isomorphic theory (including renorming) of Banach spaces |

46B20 | Geometry and structure of normed linear spaces |

46C15 | Characterizations of Hilbert spaces |

05D10 | Ramsey theory |

46B15 | Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces |

46B25 | Classical Banach spaces in the general theory |