# Methods in the Theory of Hereditarily Indecomposable Banach Spaces

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*Spiros A. Argyros; Andreas Tolias*

A general method producing Hereditarily Indecomposable (H.I.) Banach spaces is provided. We apply this method to construct a nonseparable H.I. Banach space \(Y\). This space is the dual, as well as the second dual, of a separable H.I. Banach space. Moreover the space of bounded linear operators \({\mathcal{L}}Y\) consists of elements of the form \(\lambda I+W\) where \(W\) is a weakly compact operator and hence it has separable range. Another consequence of the exhibited method is the proof of the complete dichotomy for quotients of H.I. Banach spaces. Namely we show that every separable Banach space \(Z\) not containing an isomorphic copy of \(\ell^1\) is a quotient of a separable H.I. space \(X\). Furthermore the isomorph of \(Z^*\) into \(X^*\), defined by the conjugate operator of the quotient map, is a complemented subspace of \(X^*\).

#### Table of Contents

# Table of Contents

## Methods in the Theory of Hereditarily Indecomposable Banach Spaces

- Contents v6 free
- Introduction 18 free
- Chapter 1. General results about H.I. spaces 1118 free
- Chapter 2. Schreier families and repeated averages 1522
- Chapter 3. The space X = T [G,(S[sub(nj)],1/m[sub(j)],D] and the auxiliary space Tad 2128
- Chapter 4. The basic inequality 2936
- Chapter 5. Special convex combinations in X 3946
- Chapter 6. Rapidly increasing sequences 4350
- Chapter 7. Defining D to obtain a H.I. space X[sub(G)] 4956
- Chapter 8. The predual X[sub(G)][sub(*)] of X[sub(G)]is also H.I. 5764
- Chapter 9. The structure of the space of operators L[Xsub(G)] 6168
- Chapter 10. Defining G to obtain a nonseparable H.I. space X[sup(*)][sub(G)] 6976
- Chapter 11. Complemented embedding of l[sup(p)], 1 < p < ∞, in the duals of H.I. spaces 7986
- Chapter 12. Compact families in N 8592
- Chapter 13. The space X[sub(G)] = T[G,(S[sub(ξ)],1/m[sub(j)])[sub(j)],D for an S[sub(ξ)] bounded set G 9198
- Chapter 14. Quotients of H.I. spaces 103110
- Bibliography 113120