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 \(X^*\).

Readership

Graduate students and research mathematicians interested in analysis.

Table of Contents

• Introduction
• General results about H.I. spaces
• Schreier families and repeated averages
• The space \({X= T [G,(\mathcal{S}_{n_j}, {\tfrac {1}{m_j})}_{j},D]}\) and the auxiliary space \({T_{ad}}\)
• The basic inequality
• Special convex combinations in \(X\)
• Rapidly increasing sequences
• Defining \(D\) to obtain a H.I. space \({X_G}\)
• The predual \({(X_G)_*}\) of \({X_G}\) is also H.I.
• The structure of the space of operators \({\mathcal L}(X_G)\)
• Defining \(G\) to obtain a nonseparable H.I. space \({X_G^*}\)
• Complemented embedding of \({\ell^p}, {1\le p < \infty}\), in the duals of H.I. spaces
• Compact families in \(\mathbb{N}\)
• The space \({X_{G}=T[G,(\mathcal{S}_{\xi_j},{\tfrac {1}{m_j})_{j}},D]}\) for an \({\mathcal{S}_{\xi}}\) bounded set \(G\)
• Quotients of H.I. spaces
• Bibliography