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

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.

### MSC:

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

Zbl 1160.46006
Full Text:

### References:

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