A separable Banach space X satisfies the invariant subspace property (ISP) if every bounded linear operator T ∈ L(X) admits a non-trivial closed invariant subspace. In this paper, we present the first known example of a reflexive Banach space X ISP satisfying the ISP. Moreover, this is the first known example of a Banach space satisfying the hereditary ISP, namely every infinitedimensional subspace of it satisfies the ISP. The space X ISP is hereditarily indecomposable (HI) and every operator T ∈ L(X ISP) is of the form λI + S with S a strictly singular operator. The critical property of the strictly singular operators of X ISP is that the composition of any three of them is a compact one. The construction of X ISP is based on saturation methods and it uses as an unconditional frame Tsirelson space. The new ingredient in the definition of the space is the saturation under constraints, a method initialized in a fundamental work of Edward Odell and Thomas Schlumprecht.

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Abstract. Based on a construction method introduced by J. Bourgain and F. Delbaen, we give a general definition of a Bourgain-Delbaen space and prove that every infinite dimensional separable L∞-space is isomorphic to such a space. Furthermore, we provide an example of a L∞ and asymptotic c0 space not containing c0.

Abstract. A hereditarily indecomposable Banach space Xnr is constructed that is the first known example of a L∞-space not containing c0, ℓ1, or reflexive subspaces and answers a question posed by J. Bourgain. Moreover, the space Xnr satisfies the "scalar-plus-compact" property and it is the first known space without reflexive subspaces having this property. It is constructed using the Bourgain-Delbaen method in combination with a recent version of saturation under constraints in a mixed-Tsirelson setting. As a result, the space Xnr has a shrinking finite dimensional decomposition and does not contain a boundedly complete sequence.

In this article we start a systematic study of the bi-Lipschitz geometry of lamplighter graphs. We prove that lamplighter graphs over trees bi-Lipschitzly embed into Hamming cubes with distortion at most 6. It follows that lamplighter graphs over countable trees bi-Lipschitzly embed into 1 . We study the metric behaviour of the operation of taking the lamplighter graph over the vertex-coalescence of two graphs. Based on this analysis, we provide metric characterizations of superreflexivity in terms of lamplighter graphs over star graphs or rose graphs. Finally, we show that the presence of a clique in a graph implies the presence of a Hamming cube in the lamplighter graph over it.