Abstract. A Banach space X is said to have the Daugavet property if every operator T : X → X of rank 1 satisfies Id +T = 1 + T . We show that then every weakly compact operator satisfies this equation as well and that X contains a copy of 1 . However, X need not contain a copy of L 1 . We also study pairs of spaces X ⊂ Y and operators T : X → Y satisfying J + T = 1+ T , where J : X → Y is the natural embedding. This leads to the result that a Banach space with the Daugavet property does not embed into a space with an unconditional basis. In another direction, we investigate spaces where the set of operators with Id +T = 1 + T is as small as possible and give characterisations in terms of a smoothness condition.

In this paper we show that any order continuous operator between two Riesz spaces is automatically order bounded. We also investigate different types of order convergence.

221 222 G. ANDROULAKIS ET AL. Isr. J. Math. ABSTRACT V. D. Milman proved in [20] that the product of two strictly singular operators on Lp[0, 1] (1 p < ∞) or on C[0, 1] is compact. In this note we utilize Schreier families S ξ in order to define the class of S ξ -strictly singular operators, and then we refine the technique of Milman to show that certain products of operators from this class are compact, under the assumption that the underlying Banach space has finitely many equivalence classes of Schreier-spreading sequences. Finally we define the class of S ξ -hereditarily indecomposable Banach spaces and we examine the operators on them.

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In this paper we study two types of collections of operators on a Banach space on the subject of forming operator ideals. One of the types allows us to construct an uncountable chain of closed ideals in each of the operator algebras L(ℓ 1 ⊕ ℓ q ), 1 < q < ∞, and L(ℓ 1 ⊕ c 0 ). This finishes answering a longstanding question of Pietsch.