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Banach spaces with few operators. (English) Zbl 1044.46011
Johnson, W. B. (ed.) et al., Handbook of the geometry of Banach spaces. Volume 2. Amsterdam: North-Holland (ISBN 0-444-51305-1/hbk). 1247-1297 (2003).
The focal points of this beautiful exposition are a number of longstanding open problems in the geometry of infinite-dimensional Banach spaces which were solved in the 1990’s by the author and W. T. Gowers. These include the following problems. (1) The hyperplane problem: is every (infinite-dimensional Banach space) \(X\) isomorphic to its hyperplanes? (2) The unconditional basic sequence problem: does every \(X\) contain an unconditional basic sequence? Can every \(X\) be decomposed as \(X= W\oplus Y\) where \(W\) and \(Y\) are closed infinite-dimensional subspaces? The author presents a nice short history of these and quite a few more related problems. The unifying theme of many of the problems considered is to construct a Banach space \(X\) with very few operators (e.g., any bounded linear operator \(T\) on \(X\) has the form \(T= \lambda \text{ Id } +S\) where \(S\) is strictly singular) or, more generally, with a prescribed class of operators.
The main part of the paper is devoted to the construction of hereditarily indecomposable (H. I.) Banach spaces (i.e., if \(Y\) is a subspace of \(X\) with \(Y = W\oplus Z\) then \(W\) or \(Z\) must be finite dimensional) which was first achieved by the author and W. T. Gowers. Again, the history and motivation for the construction is presented (Tsirelson’s space and Schlumprecht’s space). The author also presents a proof of the theorem of S. Argyros and V. Felouzis that \(\ell_p\) (\(1<p<\infty\)) is a quotient of an H.I. space.
This paper is a superb summary of an exciting period in Banach space theory.
For the entire collection see [Zbl 1013.46001].

46B03 Isomorphic theory (including renorming) of Banach spaces
46B20 Geometry and structure of normed linear spaces
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to functional analysis