Operators in tight by support Banach spaces

We answer the question of W.T. Gowers, giving an example of a bounded operator on a subspace of Gowers unconditional space which is not a strictly singular perturbation of a restriction of a diagonal operator. We make some observations on operators in arbitrary tight by support Banach space, showing in particular that in such space no two isomorphic infinitely dimensional subspaces form a direct sum.Comment: 17 pages. Revised version - added Remark 1.2, Theorem 2.10 and corresponding changes in the pape

Strictly singular non-compact operators on a class of HI spaces

We present a method for constructing bounded strictly singular non-compact operators on mixed Tsirelson spaces defined either by the families (A_n) or (S_n) of a certain class, as well as on spaces built on them, including hereditarily indecomposable spaces.Comment: 19 page

Primitive Zonotopes

We introduce and study a family of polytopes which can be seen as a generalization of the permutahedron of type $B_d$. We highlight connections with the largest possible diameter of the convex hull of a set of points in dimension $d$ whose coordinates are integers between $0$ and $k$, and with the computational complexity of multicriteria matroid optimization.Comment: The title was slightly modified, and the determination of the computational complexity of multicriteria matroid optimization was adde

Isomorphisms and strictly singular operators in mixed Tsirelson spaces

We study the family of isomorphisms and strictly singular operators in mixed Tsirelson spaces and their modified versions setting. We show sequential minimality of modified mixed Tsirelson spaces T_M[(\mc{S}_n,\theta_n)] satisfying some regularity conditions and present results on existence of strictly singular non-compact operators on subspaces of mixed Tsirelson spaces defined by the families (\mc{A}_n)_n and (\mc{S}_n)_n.Comment: 29 pages, no figure

Unconditionally saturated Banach space with the scalar-plus-compact property

We construct a Bourgain-Delbaen $\mathscr{L}_\infty$-space $\mathfrak{X}_{Kus}$ with strongly heterogenous structure: any bounded operator on $\mathfrak{X}_{Kus}$ is a compact perturbation of a multiple of the identity, whereas the space $\mathfrak{X}_{Kus}$ is saturated with unconditional basic sequences.Comment: The result about tightness removed,notation clarified and corrected, clarifying comments added, Journal of Functional Analysis, 201