Methods in the Theory of Hereditarily Indecomposable Banach Spaces (Memoirs of the American Mathematical Society) - Softcover

Argyros, Spiros; Tolias, Andreas

Softcover

ISBN 10: 0821835211 ISBN 13: 9780821835210

Publisher: American Mathematical Society, 2004

Synopsis

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 $X^*$.

Edition details

• Publisher: American Mathematical Society
• Publication date: 2004
• Language: English
• ISBN 10: 0821835211
• ISBN 13: 9780821835210
• Binding: Paperback
• Number of pages: 114