A hereditarily indecomposable $ {\mathcal{L}_{\infty}} $-space that solves the scalar-plus-compact problem

@article{Argyros2011AHI,
  title={A hereditarily indecomposable \$ \{\mathcal\{L\}\_\{\infty\}\} \$-space that solves the scalar-plus-compact problem},
  author={Spiros A. Argyros and Richard Haydon},
  journal={Acta Mathematica},
  year={2011},
  volume={206},
  pages={1-54}
}
We construct a hereditarily indecomposable Banach space with dual space isomorphic to ℓ1. Every bounded linear operator on this space is expressible as λI + K, with λ a scalar and K compact. 
Hereditarily indecomposable Banach algebras of diagonal operators
AbstractWe provide a characterization of the Banach spaces X with a Schauder basis (en)n∈ℕ which have the property that the dual space X* is naturally isomorphic to the space Ldiag(X) of diagonal
The strong convex property and C0-semigroups on some hereditarily indecomposable Banach spaces
In this paper, we study the strong convex compactness property on the hereditarily indecomposable Banach spaces denoted respectively by XGM and XAM constructed by T. Gowers and B. Maurey (1993) and
A dual method of constructing hereditarily indecomposable Banach spaces
A new method of defining hereditarily indecomposable Banach spaces is presented. This method provides a unified approach for constructing reflexive HI spaces and also HI spaces with no reflexive
Additive Mappings Preserving Fredholm Operators with Fixed Nullity or Defect
Let $${\cal X}$$ be an infinite-dimensional real or complex Banach space, and $${\cal B}({\cal X})$$ the Banach algebra of all bounded linear operators on $${\cal X}$$ . In this paper, given any
ON THE STRUCTURE OF SEPARABLE -SPACES
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
Operators on Bourgain–Delbaen spaces
We prove that, for a suitable choice of real numbers $a, b$, every operator from $\ell_2$ to $X_{a,b}$ and from $X_{a,b}$ to $\ell_2$ must be compact, where $X_{a,b}$ is the Bourgain- Delbaen's space.
More (\ell_r) saturated (\mathcal{L}_\infty) spaces
We present some new examples of separable (\mathcal_\infty) spaces which are (\ell_r) saturated for some (1 < r < \infty).
On the undefinability of Tsirelson's space and its descendants.
We prove that Tsirelson's space cannot be defined explicitly from the classical Banach sequence spaces. We also prove that any Banach space that is explicitly definable from a class of spaces that
The universality of ℓ1 as a dual space
Let X be a Banach space with a separable dual. We prove that X embeds isomorphically into a $${{\mathcal L}_\infty}$$ space Z whose dual is isomorphic to ℓ1. If, moreover, U is a space with separable
...
...

References

SHOWING 1-10 OF 45 REFERENCES
An indecomposable and unconditionally saturated Banach space
We construct an indecomposable reflexive Banach space $X_{ius}$ such that every infinite dimensional closed subspace contains an unconditional basic sequence. We also show that every operator $T\in
Hereditarily indecomposable, separable ℒ∞ Banach spaces with ℓ1 dual having few but not very few operators
TLDR
This construction answers a question of Argyros and Haydon ("A hereditarily indecomposable space that solves the scalar-plus-compact problem").
The dual of the bourgain-delbaen space
AbstractIt is shown that a $$\mathcal{L}_\infty $$ -space with separable dual constructed by Bourgain and Delbaen has small Szlenk index and thus does not have a quotient isomorphic toC(ωω). It
Quotient Hereditarily Indecomposable Banach Spaces
  • V. Ferenczi
  • Mathematics
    Canadian Journal of Mathematics
  • 1999
Abstract A Banach space $X$ is said to be quotient hereditarily indecomposable if no infinite dimensional quotient of a subspace of $X$ is decomposable. We provide an example of a quotient
Strictly singular non-compact operators on hereditarily indecomposable Banach spaces
An example is given of a strictly singular non-compact operator on a Hereditarily Indecomposable, reflexive, asymptotic l 1 Banach space. The construction of this operator relies on the existence of
The cofinal property of the Reflexive Indecomposable Banach spaces
It is shown that every separable reflexive Banach space is a quotient of a reflexive Hereditarily Indecomposable space, which yields that every separable reflexive Banach is isomorphic to a subspace
A construction of ℒ∞-spaces and related Banach spacesand related Banach spaces
AbstractLet λ>1. We prove that every separable Banach space E can be embedded isometrically into a separable ℒ∞λ-spaceX such thatX/E has the RNP and the Schur property. This generalizes a result in
A Remark about the Scalar-Plus-Compact Problem
In [GM] a Banach space X was constructed such that every operator from a subspace Y ⊂ X into the space is of the form λIY→X +S, where IY→X is the inclusion map and S is strictly singular. In this
...
...