Operators on asymptotic \(\ell_p\) spaces which are not compact perturbations of a multiple of the identity. (English) Zbl 1185.46005

Beanland, Kevin

Ill. J. Math. 52, No. 2, 515-532 (2008).

The paper is concerned with constructing nontrivial bounded linear operators, \(T\in \mathcal L(X)\), for certain types of separable infinite-dimensional Banach spaces \(X\). Numerous examples have been produced of spaces \(X\) such that \(\mathcal L(X) = \{\lambda I + S :\lambda\in \mathbb R\) and \(S\) is a strictly singular operator\(\}\). When this paper was written, it was unknown if an \(X\) could be found such that \(\mathcal L(X) = \{\lambda I+K :\lambda \in \mathbb R\), \(K\) is a compact operator\(\}\) (recently solved in the affirmative by S. A. Argyros and R. G. Haydon [“A hereditarily indecomposable \(L_\infty\)-space that solves the scalar-plus-compact problem” (Preprint) (2009; arXiv:0903.3921)]). Thus a motivating problem was to produce strictly singular noncompact operators on spaces \(X\) as above. The main theorem concerns spaces \(X_\mathbb N\) where \(\mathbb N\) is a norming set which is \((M,N,q)\)-Schreier. This class includes the asymptotic \(\ell_p\) hereditarily indecomposable (H.I.) spaces of [I. Deliyanni and A. Manoussakis, Ill. J. Math. 51, No.  3, 767–803 (2007; Zbl 1160.46006)].
The spaces \({\mathcal L}(X_{\mathbb N})\) are shown to admit an operator \(T\) so that
(a) \(T\) is not of the form \(\lambda I +K\), \(K\) compact.
(b) If \(X_{\mathbb N}\) is H.I., then, in addition, \(T\) is strictly singular.

Reviewer: Edward W. Odell (Austin)

MSC:

46B03	Isomorphic theory (including renorming) of Banach spaces
46B06	Asymptotic theory of Banach spaces

Keywords:

asymptotic \(\ell_p\) space; strictly singular operator; hereditarily indecomposable Banach space

Citations:

Zbl 1160.46006

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