Spaces, the Scalar-Plus-Compact Property and Invariant Subspaces

Bourgain–Delbaen L∞-spaces, the Scalar-plus-Compact property and Invariant Subspaces

Spiros A. Argyros and Richard G. Haydon

Department of Mathematics, National Technical University of Athens Mathematical Institute, University of Oxford

International Congress of Mathematicians, Rio de Janiero, August 2018

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces By combining this technique with the machinery of hereditarily indecomposable constructions, as developed by Gowers, Maurey and Argyros, the speakers were able to exhibit in 2011 a separable Banach space XK on which every bounded linear operator has the form λI + K with λ a scalar and K a compact operator. The existence of such a space had been known as the Scalar-plus-Compact Problem.

By the Aronszajn–Smith Theorem, the space XK also has the Invariant Subspace Property: that is to say every bounded linear operator on XK has a proper, non-trivial, closed invariant subspace. It is the ﬁrst space known to have this property (though, of course, whether Hilbert space does is a famous open problem).

Introduction

What we now call the “Bourgain–Delbaen construction” is a general method of constructing Banach spaces of the class L∞, inspired by a remarkable paper published in Acta Math. (1980).

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces By the Aronszajn–Smith Theorem, the space XK also has the Invariant Subspace Property: that is to say every bounded linear operator on XK has a proper, non-trivial, closed invariant subspace. It is the ﬁrst space known to have this property (though, of course, whether Hilbert space does is a famous open problem).

Introduction

What we now call the “Bourgain–Delbaen construction” is a general method of constructing Banach spaces of the class L∞, inspired by a remarkable paper published in Acta Math. (1980).

By combining this technique with the machinery of hereditarily indecomposable constructions, as developed by Gowers, Maurey and Argyros, the speakers were able to exhibit in 2011 a separable Banach space XK on which every bounded linear operator has the form λI + K with λ a scalar and K a compact operator. The existence of such a space had been known as the Scalar-plus-Compact Problem.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces (though, of course, whether Hilbert space does is a famous open problem).

Introduction

What we now call the “Bourgain–Delbaen construction” is a general method of constructing Banach spaces of the class L∞, inspired by a remarkable paper published in Acta Math. (1980).

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Introduction

What we now call the “Bourgain–Delbaen construction” is a general method of constructing Banach spaces of the class L∞, inspired by a remarkable paper published in Acta Math. (1980).

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces The BD construction has also been useful in establishing new positive results (that is to say “not just counterexamples”), notably the embedding theorem of Freeman–Odell–Schlumprecht, and more recently in ﬁnding new proofs of some important theorems of Zippin. Again, time constraints will prevent us from looking at these developments in as much detail as we should wish.

Introduction

In the last seven years, the BD method and variants of it have been used to construct a wide range of other examples: other workers involved include I. Gasparis, A. Manoussakis, M. Papadiamantis, A. Pelczar-Barwacz, P. Motakis, D. Puglisi, Th. Raikoftsalis, M. Swi˛etek,M.´ Tarbard, A. Tolias, D. Zissimopoulou. We shall not have time to look in detail at all their contributions.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Introduction

The BD construction has also been useful in establishing new positive results (that is to say “not just counterexamples”), notably the embedding theorem of Freeman–Odell–Schlumprecht, and more recently in ﬁnding new proofs of some important theorems of Zippin. Again, time constraints will prevent us from looking at these developments in as much detail as we should wish.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Plan

The plan for the rest of the talk is as follows:

L∞-spaces and the general BD-construction Standard BD-spaces The FOS embedding theorem Tsirelson spaces and “few-operators” constructions

The space XK with very few operators The Invariant Subspace Property

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Plan

The plan for the rest of the talk is as follows:

L∞-spaces and the general BD-construction Standard BD-spaces The FOS embedding theorem Tsirelson spaces and “few-operators” constructions

The space XK with very few operators The Invariant Subspace Property

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Plan

The plan for the rest of the talk is as follows:

L∞-spaces and the general BD-construction Standard BD-spaces The FOS embedding theorem Tsirelson spaces and “few-operators” constructions

The space XK with very few operators The Invariant Subspace Property

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Plan

The plan for the rest of the talk is as follows:

L∞-spaces and the general BD-construction Standard BD-spaces The FOS embedding theorem Tsirelson spaces and “few-operators” constructions

The space XK with very few operators The Invariant Subspace Property

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Plan

The plan for the rest of the talk is as follows:

L∞-spaces and the general BD-construction Standard BD-spaces The FOS embedding theorem Tsirelson spaces and “few-operators” constructions

The space XK with very few operators The Invariant Subspace Property

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Plan

The plan for the rest of the talk is as follows:

L∞-spaces and the general BD-construction Standard BD-spaces The FOS embedding theorem Tsirelson spaces and “few-operators” constructions

The space XK with very few operators The Invariant Subspace Property

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Plan

The plan for the rest of the talk is as follows:

L∞-spaces and the general BD-construction Standard BD-spaces The FOS embedding theorem Tsirelson spaces and “few-operators” constructions

The space XK with very few operators The Invariant Subspace Property

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces More precisely, a Banach space X is a L∞-space if there is a constant M such that, for every finite-dimensional subspace E of X, there is a finite-dimensional subspace F with E ⊆ F ⊆ X that is M-isomorphic to a finite-dimensional `∞-space.

So a separable space X is L∞,M iff there is an increasing sequence of ﬁnite-dimensional subspaces E ⊆ E ⊆ · · · such that S E is dense in X and 1 2 n∈N n dim En En is M-isomorphic to `∞ for each n.

We are going to investigate a particular way to construct L∞-spaces, based on successive linear extensions

L∞-spaces

A Banach space is called a L∞-space if its ﬁnite-dimensional structure looks like that of c0 (or a C (K )-space). (Lindenstrauss and Pełczynski´ 1968)

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces So a separable space X is L∞,M iff there is an increasing sequence of ﬁnite-dimensional subspaces E ⊆ E ⊆ · · · such that S E is dense in X and 1 2 n∈N n dim En En is M-isomorphic to `∞ for each n.

We are going to investigate a particular way to construct L∞-spaces, based on successive linear extensions

L∞-spaces

A Banach space is called a L∞-space if its ﬁnite-dimensional structure looks like that of c0 (or a C (K )-space). (Lindenstrauss and Pełczynski´ 1968)

More precisely, a Banach space X is a L∞-space if there is a constant M such that, for every finite-dimensional subspace E of X, there is a finite-dimensional subspace F with E ⊆ F ⊆ X that is M-isomorphic to a finite-dimensional `∞-space.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces We are going to investigate a particular way to construct L∞-spaces, based on successive linear extensions

L∞-spaces

A Banach space is called a L∞-space if its ﬁnite-dimensional structure looks like that of c0 (or a C (K )-space). (Lindenstrauss and Pełczynski´ 1968)

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces L∞-spaces

A Banach space is called a L∞-space if its ﬁnite-dimensional structure looks like that of c0 (or a C (K )-space). (Lindenstrauss and Pełczynski´ 1968)

We are going to investigate a particular way to construct L∞-spaces, based on successive linear extensions

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces (Γ) Γ We write R for the subspace of R consisting of functions of ﬁnite support. With (Γ) S F the above convention, R = F⊂Γ,F ﬁnite R .

Γ (Γ) There is a natural duality between the spaces R and R , given by P Γ hf , xi = γ∈Γ f (γ)x(γ), and generally we shall think of elements of R as vectors, (Γ) while elements of R are seen as functionals. For clarity (we hope!) functionals will be given names adorned with a star, so that (Γ) ∗ the typical element of R will be denoted f , rather than f as above. In particular, the function that takes the value 1 at γ, while vanishing elsewhere, ∗ will be denoted eγ if we think of it as the usual unit vector, and eγ if we think of it as the evaluation functional x 7→ x(γ).

A little notation

Γ We write R for the vector space of all real-valued functions on a set Γ, and when Γ F ⊂ Γ we identify RF with the subspace of R consisting of functions that vanish off the subset F.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Γ (Γ) There is a natural duality between the spaces R and R , given by P Γ hf , xi = γ∈Γ f (γ)x(γ), and generally we shall think of elements of R as vectors, (Γ) while elements of R are seen as functionals. For clarity (we hope!) functionals will be given names adorned with a star, so that (Γ) ∗ the typical element of R will be denoted f , rather than f as above. In particular, the function that takes the value 1 at γ, while vanishing elsewhere, ∗ will be denoted eγ if we think of it as the usual unit vector, and eγ if we think of it as the evaluation functional x 7→ x(γ).

A little notation

Γ We write R for the vector space of all real-valued functions on a set Γ, and when Γ F ⊂ Γ we identify RF with the subspace of R consisting of functions that vanish off the subset F.

(Γ) Γ We write R for the subspace of R consisting of functions of ﬁnite support. With (Γ) S F the above convention, R = F⊂Γ,F ﬁnite R .

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces For clarity (we hope!) functionals will be given names adorned with a star, so that (Γ) ∗ the typical element of R will be denoted f , rather than f as above. In particular, the function that takes the value 1 at γ, while vanishing elsewhere, ∗ will be denoted eγ if we think of it as the usual unit vector, and eγ if we think of it as the evaluation functional x 7→ x(γ).

A little notation

Γ We write R for the vector space of all real-valued functions on a set Γ, and when Γ F ⊂ Γ we identify RF with the subspace of R consisting of functions that vanish off the subset F.

(Γ) Γ We write R for the subspace of R consisting of functions of ﬁnite support. With (Γ) S F the above convention, R = F⊂Γ,F ﬁnite R .

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces In particular, the function that takes the value 1 at γ, while vanishing elsewhere, ∗ will be denoted eγ if we think of it as the usual unit vector, and eγ if we think of it as the evaluation functional x 7→ x(γ).

A little notation

Γ We write R for the vector space of all real-valued functions on a set Γ, and when Γ F ⊂ Γ we identify RF with the subspace of R consisting of functions that vanish off the subset F.

(Γ) Γ We write R for the subspace of R consisting of functions of ﬁnite support. With (Γ) S F the above convention, R = F⊂Γ,F ﬁnite R .

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces A little notation

Γ We write R for the vector space of all real-valued functions on a set Γ, and when Γ F ⊂ Γ we identify RF with the subspace of R consisting of functions that vanish off the subset F.

(Γ) Γ We write R for the subspace of R consisting of functions of ﬁnite support. With (Γ) S F the above convention, R = F⊂Γ,F ﬁnite R .

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Γn Γ We can deﬁne, for each n, a linear extension mapping in+1,n : R → R n+1 , by ( u(γ) if γ ∈ Γ (γ) = n in+1,nu ∗ hcγ, ui if γ ∈ ∆n+1.

By forming compositions im,m−1 ◦ im−1,m−2 ◦ · · · ◦ in+1,n and letting m increase, we Γn Γ obtain, for each n, a linear extension mapping in : R → R . Writing rn for the Γ Γn restriction mapping R → R , we have the extension condition rninrn = rn and the compatibility condition imrmin = in for n < m.

Γn The images Fn = in[R ] form an increasing sequence of ﬁnite-dimensional Γ Γ subspaces of R . For each n, Pn = inrn is a projection on R with image Fn.

Successive extension mappings (easy linear algebra)

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces By forming compositions im,m−1 ◦ im−1,m−2 ◦ · · · ◦ in+1,n and letting m increase, we Γn Γ obtain, for each n, a linear extension mapping in : R → R . Writing rn for the Γ Γn restriction mapping R → R , we have the extension condition rninrn = rn and the compatibility condition imrmin = in for n < m.

Γn The images Fn = in[R ] form an increasing sequence of ﬁnite-dimensional Γ Γ subspaces of R . For each n, Pn = inrn is a projection on R with image Fn.

Successive extension mappings (easy linear algebra)

Let (Γ ) be an increasing sequence of ﬁnite sets with union Γ = S Γ , and n n∈N n∈N n introduce the difference sets ∆1 = Γ1, ∆n+1 = Γn+1 \ Γn. Suppose that, for each n ∗ Γn ∗ and each γ ∈ ∆n+1, we specify some cγ ∈ R . For completeness, set cγ = 0 when γ ∈ Γ1. Γn Γ We can deﬁne, for each n, a linear extension mapping in+1,n : R → R n+1 , by ( u(γ) if γ ∈ Γ (γ) = n in+1,nu ∗ hcγ, ui if γ ∈ ∆n+1.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Γn The images Fn = in[R ] form an increasing sequence of ﬁnite-dimensional Γ Γ subspaces of R . For each n, Pn = inrn is a projection on R with image Fn.

Successive extension mappings (easy linear algebra)

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Successive extension mappings (easy linear algebra)

Γn The images Fn = in[R ] form an increasing sequence of ﬁnite-dimensional Γ Γ subspaces of R . For each n, Pn = inrn is a projection on R with image Fn.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Left- and right-triangular bases

∗ Continuing from the last slide, we introduce vectors dγ and functionals dγ deﬁned as follows

dγ = ineγ when γ ∈ ∆n ∗ ∗ ∗ dγ = eγ − cγ

Observation ∗ (Γ) (dγ )γ∈Γ is an algebraic basis of R Γ ∗ (dγ)γ∈Γ is the unique system in R that is biorthogonal to (dγ )γ∈Γ, and is an S algebraic basis of F = n Fn the projection P = i r : Γ → F is given by P x = P hd ∗, xid n n n R n n γ∈Γn γ γ ∗ (Γ) Γn the dual projection Pn takes R onto R and is given by P∗f ∗ = P hf ∗, d id ∗ n γ∈Γn γ γ

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces A lot of good things happen.

For each n, Fn is M-isomorphic to `∞(Γn), and hence the subspace X of ` (Γ), deﬁned as X = S F , is ∞ n∈N n L∞,M ∗ ∗ ∗ The functionals dγ = eγ − cγ form a Schauder basis of `1(Γ) The biorthogonal vectors dγ ∈ `∞(Γ) form a Schauder basis of X ∗ The projections Pn : `∞(Γ) → Fn and Pn : `1(Γ) → `1(Γn) have norm at most M.

The general BD construction

Suppose that in the preceding discussion the extension operators in turn out to take values in `∞(Γ) with a uniform bound kink ≤ M for the norm of in as an operator from `∞(Γn) to `∞(Γ).

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces For each n, Fn is M-isomorphic to `∞(Γn), and hence the subspace X of ` (Γ), deﬁned as X = S F , is ∞ n∈N n L∞,M ∗ ∗ ∗ The functionals dγ = eγ − cγ form a Schauder basis of `1(Γ) The biorthogonal vectors dγ ∈ `∞(Γ) form a Schauder basis of X ∗ The projections Pn : `∞(Γ) → Fn and Pn : `1(Γ) → `1(Γn) have norm at most M.

The general BD construction

Suppose that in the preceding discussion the extension operators in turn out to take values in `∞(Γ) with a uniform bound kink ≤ M for the norm of in as an operator from `∞(Γn) to `∞(Γ). A lot of good things happen.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces The general BD construction

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Although the use of extension operators might seem to limit the scope of constructions of this kind, Argyros, Gasparis and Motakis recently showed that every separable L∞-space (and, in particular, every isomorphic predual of `1) is isomorphic to a BD-space.

∗ What is missing at this level of generality is a useful condition on the functionals cγ that would guarantee uniform boundedness of the extensions in. It would also be nice to have a criterion for the canonically embedded isomorphic copy of `1(Γ) to make up the whole of X ∗.

The problems of boundedness and duality

What we have just described is the General BD Construction and the resulting space X is what we call a (general) BD-space. The set Γ, equipped with the compatible sequence of extension mappings (in), or equivalently with the family of ∗ functionals (cγ), is what we call a BD-set.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces ∗ What is missing at this level of generality is a useful condition on the functionals cγ that would guarantee uniform boundedness of the extensions in. It would also be nice to have a criterion for the canonically embedded isomorphic copy of `1(Γ) to make up the whole of X ∗.

The problems of boundedness and duality

Although the use of extension operators might seem to limit the scope of constructions of this kind, Argyros, Gasparis and Motakis recently showed that every separable L∞-space (and, in particular, every isomorphic predual of `1) is isomorphic to a BD-space.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces The problems of boundedness and duality

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Plan

L∞-spaces and the general BD-construction Standard BD-spaces The FOS embedding theorem Tsirelson spaces and “few-operators” constructions

The space XK with very few operators The Invariant Subspace Property

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces We say that Γ is a Standard BD-set with weight β if, for all n and all γ ∈ ∆n+1, the ∗ functional cγ has one of the forms

∗ ∗ ∗ cγ = β(I − Ps )b

∗ with 0 ≤ s ≤ n and b ∈ ball `1(Γn \ Γs), or

∗ ∗ ∗ ∗ cγ = eξ +β(I − Ps )b

∗ with 1 ≤ s ≤ n, ξ ∈ Γs and b ∈ ball `1(Γn \ Γs).

Standard BD-sets

The original paper of Bourgain and Delbaen gave a special way to deﬁne ∗ functionals cγ that guarantees the uniform boundedness of the extension operators in. We turn this into a deﬁnition.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Standard BD-sets

The original paper of Bourgain and Delbaen gave a special way to deﬁne ∗ functionals cγ that guarantees the uniform boundedness of the extension operators in. We turn this into a deﬁnition.

We say that Γ is a Standard BD-set with weight β if, for all n and all γ ∈ ∆n+1, the ∗ functional cγ has one of the forms

∗ ∗ ∗ cγ = β(I − Ps )b

∗ with 0 ≤ s ≤ n and b ∈ ball `1(Γn \ Γs), or

∗ ∗ ∗ ∗ cγ = eξ +β(I − Ps )b

∗ with 1 ≤ s ≤ n, ξ ∈ Γs and b ∈ ball `1(Γn \ Γs).

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces We then call X(Γ) a standard BD-space.

Boundedness for standard BD-spaces

A crucial but easy estimate essentially due to B and D

Let Γ be a standard BD set of weight β. Then the extension operators in are −1 uniformly norm bounded by M = (1 − 2β) and X(Γ) is a L∞,M -space.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Boundedness for standard BD-spaces

A crucial but easy estimate essentially due to B and D

Let Γ be a standard BD set of weight β. Then the extension operators in are −1 uniformly norm bounded by M = (1 − 2β) and X(Γ) is a L∞,M -space.

We then call X(Γ) a standard BD-space.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces So to define a standard BD-structure on a set Γ we need first to express it as an increasing union of finite subsets Γn and then to define functions

base :Γ → Γ ∪ {undeﬁned}, top :Γ → ball `1(Γ), cut :Γ → N

satisfying the obvious constraints.

We still have freedom to specify the weight β and it should be emphasized that different choices of β can lead to Banach spaces of very different behaviour. There is even a degenerate case that is more useful than it might appear: we can set β = 0.

Standard terminology for standard BD-spaces

∗ ∗ ∗ ∗ ∗ ∗ Recall that cγ is to be either β(I − Ps )b or eξ + β(I − Ps )b . We call ξ (when it exists) the base of γ, b∗ the top and s the cut.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces We still have freedom to specify the weight β and it should be emphasized that different choices of β can lead to Banach spaces of very different behaviour. There is even a degenerate case that is more useful than it might appear: we can set β = 0.

Standard terminology for standard BD-spaces

∗ ∗ ∗ ∗ ∗ ∗ Recall that cγ is to be either β(I − Ps )b or eξ + β(I − Ps )b . We call ξ (when it exists) the base of γ, b∗ the top and s the cut.

So to define a standard BD-structure on a set Γ we need first to express it as an increasing union of finite subsets Γn and then to define functions

base :Γ → Γ ∪ {undeﬁned}, top :Γ → ball `1(Γ), cut :Γ → N

satisfying the obvious constraints.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces There is even a degenerate case that is more useful than it might appear: we can set β = 0.

Standard terminology for standard BD-spaces

∗ ∗ ∗ ∗ ∗ ∗ Recall that cγ is to be either β(I − Ps )b or eξ + β(I − Ps )b . We call ξ (when it exists) the base of γ, b∗ the top and s the cut.

So to define a standard BD-structure on a set Γ we need first to express it as an increasing union of finite subsets Γn and then to define functions

base :Γ → Γ ∪ {undeﬁned}, top :Γ → ball `1(Γ), cut :Γ → N

satisfying the obvious constraints.

We still have freedom to specify the weight β and it should be emphasized that different choices of β can lead to Banach spaces of very different behaviour.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Standard terminology for standard BD-spaces

∗ ∗ ∗ ∗ ∗ ∗ Recall that cγ is to be either β(I − Ps )b or eξ + β(I − Ps )b . We call ξ (when it exists) the base of γ, b∗ the top and s the cut.

So to define a standard BD-structure on a set Γ we need first to express it as an increasing union of finite subsets Γn and then to define functions

base :Γ → Γ ∪ {undeﬁned}, top :Γ → ball `1(Γ), cut :Γ → N

satisfying the obvious constraints.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Theorem ∗ Let Γ be a standard BD-set. Then the dual X(Γ) is naturally identiﬁable with `1(Γ) if and only if there is no inﬁnite sequence (ξn) in Γ with ξn = base ξn+1 for all n, that is to say, if and only if the tree structure on Γ is well-founded.

When Γ is well-founded, it has a countable ordinal index that becomes important when we look at the ﬁner structure of X(Γ).

Duality for standard BD-spaces

A standard BD-set Γ has a tree-structure in which the immediate successors of ξ are the elements γ with base γ = ξ. This is intimately related to duality issues.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces When Γ is well-founded, it has a countable ordinal index that becomes important when we look at the ﬁner structure of X(Γ).

Duality for standard BD-spaces

A standard BD-set Γ has a tree-structure in which the immediate successors of ξ are the elements γ with base γ = ξ. This is intimately related to duality issues.

Theorem ∗ Let Γ be a standard BD-set. Then the dual X(Γ) is naturally identiﬁable with `1(Γ) if and only if there is no inﬁnite sequence (ξn) in Γ with ξn = base ξn+1 for all n, that is to say, if and only if the tree structure on Γ is well-founded.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Duality for standard BD-spaces

A standard BD-set Γ has a tree-structure in which the immediate successors of ξ are the elements γ with base γ = ξ. This is intimately related to duality issues.

When Γ is well-founded, it has a countable ordinal index that becomes important when we look at the ﬁner structure of X(Γ).

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Plan

L∞-spaces and the general BD-construction Standard BD-spaces The FOS embedding theorem Tsirelson spaces and “few-operators” constructions

The space XK with very few operators The Invariant Subspace Property

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Theorem [FOS (2011)] ∗ 1 Let Y be a Banach space with separable dual Y . Then, for each 0 < < 2 , there exists a well-founded special BD-set Γ and a (1 + )-isomorphic embedding Y → X(Γ).

One approach to proving this theorem involves a careful construction of a certain -net Γ in ball Y ∗, which we then equip with a standard BD structure in such that the natural map J : Y → `∞(Γ); (Jy)(γ) = γ(y) takes values in X(Γ).

The embedding theorem of Freeman, Odell and Schlumprecht

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces One approach to proving this theorem involves a careful construction of a certain -net Γ in ball Y ∗, which we then equip with a standard BD structure in such that the natural map J : Y → `∞(Γ); (Jy)(γ) = γ(y) takes values in X(Γ).

The embedding theorem of Freeman, Odell and Schlumprecht

It is an elementary fact that every separable Banach space is a quotient of `1. The surprising theorem of FOS is that when Y ∗ is a separable dual space then the quotient operator can itself be taken to be the dual of an embedding Y → X where X is an isomorphic predual of `1. Theorem [FOS (2011)] ∗ 1 Let Y be a Banach space with separable dual Y . Then, for each 0 < < 2 , there exists a well-founded special BD-set Γ and a (1 + )-isomorphic embedding Y → X(Γ).

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces The embedding theorem of Freeman, Odell and Schlumprecht

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Plan

L∞-spaces and the general BD-construction Standard BD-spaces The FOS embedding theorem Tsirelson spaces and “few-operators” constructions

The space XK with very few operators The Invariant Subspace Property

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces ( ) He deﬁned a norm on R N by

∗ kxkT = sup |hf , xi|, f ∗∈W

where the norming set W ⊂ R(N) is taken to be the smallest convex set such that ∗ ±en ∈ W for all n, and

1 ∗ ∗ ∗ ∗ ∗ 2 (f1 + f2 + ··· + fn ) ∈ W whenever f1 ,..., fn ∈ W , ∗ ∗ max supp fk < min supp fk+1 (1 ≤ k < n) ∗ and n ≤ min supp f1 .

( ) The space T is the completion of R N for this norm.

Tsirelson spaces

In 1974, Tsirelson constructed his famous space T that contains no isomorphic copy of c0, nor of any `p.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces 1 ∗ ∗ ∗ ∗ ∗ 2 (f1 + f2 + ··· + fn ) ∈ W whenever f1 ,..., fn ∈ W , ∗ ∗ max supp fk < min supp fk+1 (1 ≤ k < n) ∗ and n ≤ min supp f1 .

( ) The space T is the completion of R N for this norm.

Tsirelson spaces

In 1974, Tsirelson constructed his famous space T that contains no isomorphic copy of c0, nor of any `p.

( ) He deﬁned a norm on R N by

∗ kxkT = sup |hf , xi|, f ∗∈W

where the norming set W ⊂ R(N) is taken to be the smallest convex set such that ∗ ±en ∈ W for all n, and

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Tsirelson spaces

In 1974, Tsirelson constructed his famous space T that contains no isomorphic copy of c0, nor of any `p.

( ) He deﬁned a norm on R N by

∗ kxkT = sup |hf , xi|, f ∗∈W

where the norming set W ⊂ R(N) is taken to be the smallest convex set such that ∗ ±en ∈ W for all n, and

1 ∗ ∗ ∗ ∗ ∗ 2 (f1 + f2 + ··· + fn ) ∈ W whenever f1 ,..., fn ∈ W , ∗ ∗ max supp fk < min supp fk+1 (1 ≤ k < n) ∗ and n ≤ min supp f1 .

( ) The space T is the completion of R N for this norm.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces and less obviously by replacing the ∗ condition n ≤ min supp f1 with

∗ ∗ {min supp f1 ,..., min supp fn } ∈ M ,

for a suitable (regular) family M of ﬁnite subsets of N.

We can even allow a sequence (βi )i∈N of distinct weights, with associated regular families Mi , obtaining a Mixed Tsirelson space.

Schlumprecht’s arbitrarily distortable space S (1991) is a space of this type, and its discovery was a necessary step towards the celebrated counterexamples of Gowers (1993-4) and the Gowers–Maurey theory of hereditarily indecomposable (HI) Banach spaces.

Tsirelson spaces – generalized and mixed

1 We can obviously generalize Tsirelson’s construction by replacing the constant 2 with an arbitrary positive weight β < 1,

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces We can even allow a sequence (βi )i∈N of distinct weights, with associated regular families Mi , obtaining a Mixed Tsirelson space.

Tsirelson spaces – generalized and mixed

1 We can obviously generalize Tsirelson’s construction by replacing the constant 2 with an arbitrary positive weight β < 1, and less obviously by replacing the ∗ condition n ≤ min supp f1 with

∗ ∗ {min supp f1 ,..., min supp fn } ∈ M ,

for a suitable (regular) family M of ﬁnite subsets of N.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Schlumprecht’s arbitrarily distortable space S (1991) is a space of this type, and its discovery was a necessary step towards the celebrated counterexamples of Gowers (1993-4) and the Gowers–Maurey theory of hereditarily indecomposable (HI) Banach spaces.

Tsirelson spaces – generalized and mixed

1 We can obviously generalize Tsirelson’s construction by replacing the constant 2 with an arbitrary positive weight β < 1, and less obviously by replacing the ∗ condition n ≤ min supp f1 with

∗ ∗ {min supp f1 ,..., min supp fn } ∈ M ,

for a suitable (regular) family M of ﬁnite subsets of N.

We can even allow a sequence (βi )i∈N of distinct weights, with associated regular families Mi , obtaining a Mixed Tsirelson space.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Tsirelson spaces – generalized and mixed

1 We can obviously generalize Tsirelson’s construction by replacing the constant 2 with an arbitrary positive weight β < 1, and less obviously by replacing the ∗ condition n ≤ min supp f1 with

∗ ∗ {min supp f1 ,..., min supp fn } ∈ M ,

for a suitable (regular) family M of ﬁnite subsets of N.

We can even allow a sequence (βi )i∈N of distinct weights, with associated regular families Mi , obtaining a Mixed Tsirelson space.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Typically, we take a sequence of weights βi tending rapidly to 0, for instance −2i βi = 2 , and a rapidly increasing sequence of regular families Mi . The resulting space is denoted T [(βi , Mi )i∈N] which we abbreviate to Tmixed.

It is an important fact about the original Tsirelson space T that every normalized block sequence of vectors in T is equivalent to a subsequence of the unit vector basis (en) in T .

This is not the case for mixed Tsirelson spaces, though typically there is a class of special sequences (called RIS) for which Tmixed-upper estimates hold. This class is rich in the sense that every inﬁnite-dimensional subspace contains a RIS.

Mixed Tsirelson spaces and RIS

The general deﬁnition of a mixed Tsirelson space is due to Argyros and Deliyanni (1992) and its ﬂexibility has been essential for further developments over the last 20 years.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces It is an important fact about the original Tsirelson space T that every normalized block sequence of vectors in T is equivalent to a subsequence of the unit vector basis (en) in T .

Mixed Tsirelson spaces and RIS

The general deﬁnition of a mixed Tsirelson space is due to Argyros and Deliyanni (1992) and its ﬂexibility has been essential for further developments over the last 20 years.

Typically, we take a sequence of weights βi tending rapidly to 0, for instance −2i βi = 2 , and a rapidly increasing sequence of regular families Mi . The resulting space is denoted T [(βi , Mi )i∈N] which we abbreviate to Tmixed.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces This is not the case for mixed Tsirelson spaces, though typically there is a class of special sequences (called RIS) for which Tmixed-upper estimates hold. This class is rich in the sense that every inﬁnite-dimensional subspace contains a RIS.

Mixed Tsirelson spaces and RIS

The general deﬁnition of a mixed Tsirelson space is due to Argyros and Deliyanni (1992) and its ﬂexibility has been essential for further developments over the last 20 years.

It is an important fact about the original Tsirelson space T that every normalized block sequence of vectors in T is equivalent to a subsequence of the unit vector basis (en) in T .

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Mixed Tsirelson spaces and RIS

The general deﬁnition of a mixed Tsirelson space is due to Argyros and Deliyanni (1992) and its ﬂexibility has been essential for further developments over the last 20 years.

It is an important fact about the original Tsirelson space T that every normalized block sequence of vectors in T is equivalent to a subsequence of the unit vector basis (en) in T .

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces A typical construction of a space with few operators would start with the

ingredients (βi , Mi )i∈N of a mixed Tsirelson space, but add additional restrictions ∗ on the functionals fk that are allowed in the creation of new elements

∗ ∗ ∗ βi (f1 + f2 + ··· + fn )

of the norming set W .

Using a coding method that goes back to Maurey and Rosenthal (1977) this can be done in such a way that for every operator T on the resulting Banach space X there is a scalar λ such that kTxn − λxnk → 0 for every RIS (xn). Since every inﬁnite-dimensional subspace contains a RIS, T − λI is strictly singular.

Few operators

A Banach space X is said to have few operators if every bounded linear operator T on X has the form λI + S, with S strictly singular (i.e. there is no inﬁnite dimensional subspace on which S is an isomorphism).

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Using a coding method that goes back to Maurey and Rosenthal (1977) this can be done in such a way that for every operator T on the resulting Banach space X there is a scalar λ such that kTxn − λxnk → 0 for every RIS (xn). Since every inﬁnite-dimensional subspace contains a RIS, T − λI is strictly singular.

Few operators

A typical construction of a space with few operators would start with the

ingredients (βi , Mi )i∈N of a mixed Tsirelson space, but add additional restrictions ∗ on the functionals fk that are allowed in the creation of new elements

∗ ∗ ∗ βi (f1 + f2 + ··· + fn )

of the norming set W .

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Since every inﬁnite-dimensional subspace contains a RIS, T − λI is strictly singular.

Few operators

A typical construction of a space with few operators would start with the

ingredients (βi , Mi )i∈N of a mixed Tsirelson space, but add additional restrictions ∗ on the functionals fk that are allowed in the creation of new elements

∗ ∗ ∗ βi (f1 + f2 + ··· + fn )

of the norming set W .

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Few operators

A typical construction of a space with few operators would start with the

ingredients (βi , Mi )i∈N of a mixed Tsirelson space, but add additional restrictions ∗ on the functionals fk that are allowed in the creation of new elements

∗ ∗ ∗ βi (f1 + f2 + ··· + fn )

of the norming set W .

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Androulakis and Schlumprecht (2001) showed that the Gowers–Maurey space does not have very few operators by constructing an operator that is strictly singular and non-compact; the same seems to be true of other spaces constructed by mixed-Tsirelson techniques alone.

Our quest for a space with very few operators is going to bring us back to the BD-construction, and the ﬁrst step will be to connect standard BD-spaces to Tsirelson norms.

Very few operators

We say that a Banach space has very few operators (or that X has the Scalar plus Compact Property) if every T can be written T = λI + K with K compact.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Our quest for a space with very few operators is going to bring us back to the BD-construction, and the ﬁrst step will be to connect standard BD-spaces to Tsirelson norms.

Very few operators

We say that a Banach space has very few operators (or that X has the Scalar plus Compact Property) if every T can be written T = λI + K with K compact.

Androulakis and Schlumprecht (2001) showed that the Gowers–Maurey space does not have very few operators by constructing an operator that is strictly singular and non-compact; the same seems to be true of other spaces constructed by mixed-Tsirelson techniques alone.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Very few operators

We say that a Banach space has very few operators (or that X has the Scalar plus Compact Property) if every T can be written T = λI + K with K compact.

Our quest for a space with very few operators is going to bring us back to the BD-construction, and the ﬁrst step will be to connect standard BD-spaces to Tsirelson norms.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Plan

L∞-spaces and the general BD-construction Standard BD-spaces The FOS embedding theorem Tsirelson spaces and “few-operators” constructions

The space XK with very few operators The Invariant Subspace Property

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces ∗ ∗ ∗ Since dγ = eγ − cγ we can rewrite this as

∗ ∗ ∗ ∗ ∗ eγ = eξ + β(I − Ps )b + dγ .

∗ We can do the same with eξ , and continue until (as we must) we reach an element with no base. What we end up with is the following:

a X e∗ = β(I − P∗ )b∗ + d ∗ γ sj j ξj j=1

where ξa = γ and ξj = base ξj+1 for j < a. We call this identity the Evaluation ∗ Analysis of γ. Its importance is that it expresses the evaluation eγ in terms that are ∗ ∗ well adapted to the basis (dξ ) of `1(Γ) = X(Γ) .

The evaluation analysis

Let Γ be a well-founded standard BD set of weight β and let γ be an element of Γ. ∗ ∗ ∗ ∗ Recall that cγ = eξ + β(I − Ps )b if ξ = base γ is deﬁned.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces ∗ We can do the same with eξ , and continue until (as we must) we reach an element with no base. What we end up with is the following:

a X e∗ = β(I − P∗ )b∗ + d ∗ γ sj j ξj j=1

The evaluation analysis

Let Γ be a well-founded standard BD set of weight β and let γ be an element of Γ. ∗ ∗ ∗ ∗ Recall that cγ = eξ + β(I − Ps )b if ξ = base γ is deﬁned.

∗ ∗ ∗ Since dγ = eγ − cγ we can rewrite this as

∗ ∗ ∗ ∗ ∗ eγ = eξ + β(I − Ps )b + dγ .

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces a X e∗ = β(I − P∗ )b∗ + d ∗ γ sj j ξj j=1

The evaluation analysis

Let Γ be a well-founded standard BD set of weight β and let γ be an element of Γ. ∗ ∗ ∗ ∗ Recall that cγ = eξ + β(I − Ps )b if ξ = base γ is deﬁned.

∗ ∗ ∗ Since dγ = eγ − cγ we can rewrite this as

∗ ∗ ∗ ∗ ∗ eγ = eξ + β(I − Ps )b + dγ .

∗ We can do the same with eξ , and continue until (as we must) we reach an element with no base. What we end up with is the following:

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces ∗ Its importance is that it expresses the evaluation eγ in terms that are ∗ ∗ well adapted to the basis (dξ ) of `1(Γ) = X(Γ) .

The evaluation analysis

Let Γ be a well-founded standard BD set of weight β and let γ be an element of Γ. ∗ ∗ ∗ ∗ Recall that cγ = eξ + β(I − Ps )b if ξ = base γ is deﬁned.

∗ ∗ ∗ Since dγ = eγ − cγ we can rewrite this as

∗ ∗ ∗ ∗ ∗ eγ = eξ + β(I − Ps )b + dγ .

∗ We can do the same with eξ , and continue until (as we must) we reach an element with no base. What we end up with is the following:

a X e∗ = β(I − P∗ )b∗ + d ∗ γ sj j ξj j=1

where ξa = γ and ξj = base ξj+1 for j < a. We call this identity the Evaluation Analysis of γ.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces The evaluation analysis

Let Γ be a well-founded standard BD set of weight β and let γ be an element of Γ. ∗ ∗ ∗ ∗ Recall that cγ = eξ + β(I − Ps )b if ξ = base γ is deﬁned.

∗ ∗ ∗ Since dγ = eγ − cγ we can rewrite this as

∗ ∗ ∗ ∗ ∗ eγ = eξ + β(I − Ps )b + dγ .

∗ We can do the same with eξ , and continue until (as we must) we reach an element with no base. What we end up with is the following:

a X e∗ = β(I − P∗ )b∗ + d ∗ γ sj j ξj j=1

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces The presence of the sum β Pa (I − P∗ )b∗ obviously puts us in mind of Tsirelson j=1 sj j norms, and with effort we can prove the following. First Basic Inequality Let Γ be a well-founded, standard BD-set of weight β. Let M be a regular family that contains all histories of elements of Γ. Then arbitrary normalized block sequences in X(Γ) admit T (β, M )-upper estimates.

I have cheated slightly: sometimes M needs to be slightly bigger than this.

A basic inequality

In the evaluation analysis

a X e∗ = β(I − P∗ )b∗ + d ∗ γ sj j ξj j=1

we call a the age of γ, and {s1,..., sa} its history.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces First Basic Inequality Let Γ be a well-founded, standard BD-set of weight β. Let M be a regular family that contains all histories of elements of Γ. Then arbitrary normalized block sequences in X(Γ) admit T (β, M )-upper estimates.

I have cheated slightly: sometimes M needs to be slightly bigger than this.

A basic inequality

In the evaluation analysis

a X e∗ = β(I − P∗ )b∗ + d ∗ γ sj j ξj j=1

we call a the age of γ, and {s1,..., sa} its history.

The presence of the sum β Pa (I − P∗ )b∗ obviously puts us in mind of Tsirelson j=1 sj j norms, and with effort we can prove the following.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces I have cheated slightly: sometimes M needs to be slightly bigger than this.

A basic inequality

In the evaluation analysis

a X e∗ = β(I − P∗ )b∗ + d ∗ γ sj j ξj j=1

we call a the age of γ, and {s1,..., sa} its history.

The presence of the sum β Pa (I − P∗ )b∗ obviously puts us in mind of Tsirelson j=1 sj j norms, and with effort we can prove the following. First Basic Inequality Let Γ be a well-founded, standard BD-set of weight β. Let M be a regular family that contains all histories of elements of Γ. Then arbitrary normalized block sequences in X(Γ) admit T (β, M )-upper estimates.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces A basic inequality

In the evaluation analysis

a X e∗ = β(I − P∗ )b∗ + d ∗ γ sj j ξj j=1

we call a the age of γ, and {s1,..., sa} its history.

I have cheated slightly: sometimes M needs to be slightly bigger than this.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces We then take Mi to be a regular family that contains the histories of all γ of weight βi . We get a Tmixed-upper estimate for a suitably deﬁned class of sequences in X(Γ) (as always, referred to as RIS) At this point, we do not seem to have advanced much from where we were before, since our upper estimates still apply only to some special sorts of sequences. But the local `∞-structure comes to the rescue, enabling us to prove a crucial lemma. Crucial Lemma Let Γ be a well-founded standard BD-space with mixed weights, let Z be any Banach space and let S : X(Γ) → Z be a bounded linear operator. If kSxnk → 0 for every RIS then kSxnk → 0 for every bounded, weakly null sequence, and hence S is compact.

BD-sets with mixed weights

The next step is to generalize our notion of standard BD-space to allow different elements to have different weights, drawn from a suitable sequence (βi ) but subject to the condition weight γ = weight base γ when this makes sense.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces At this point, we do not seem to have advanced much from where we were before, since our upper estimates still apply only to some special sorts of sequences. But the local `∞-structure comes to the rescue, enabling us to prove a crucial lemma. Crucial Lemma Let Γ be a well-founded standard BD-space with mixed weights, let Z be any Banach space and let S : X(Γ) → Z be a bounded linear operator. If kSxnk → 0 for every RIS then kSxnk → 0 for every bounded, weakly null sequence, and hence S is compact.

BD-sets with mixed weights

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces But the local `∞-structure comes to the rescue, enabling us to prove a crucial lemma. Crucial Lemma Let Γ be a well-founded standard BD-space with mixed weights, let Z be any Banach space and let S : X(Γ) → Z be a bounded linear operator. If kSxnk → 0 for every RIS then kSxnk → 0 for every bounded, weakly null sequence, and hence S is compact.

BD-sets with mixed weights

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Crucial Lemma Let Γ be a well-founded standard BD-space with mixed weights, let Z be any Banach space and let S : X(Γ) → Z be a bounded linear operator. If kSxnk → 0 for every RIS then kSxnk → 0 for every bounded, weakly null sequence, and hence S is compact.

BD-sets with mixed weights

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces BD-sets with mixed weights

We then take Mi to be a regular family that contains the histories of all γ of weight βi . We get a Tmixed-upper estimate for a suitably deﬁned class of sequences in X(Γ) (as always, referred to as RIS) At this point, we do not seem to have advanced much from where we were before, since our upper estimates still apply only to some special sorts of sequences. But the local `∞-structure comes to the rescue, enabling us to prove a crucial lemma. Crucial Lemma Let Γ be a well-founded standard BD-space with mixed weights, let Z be any Banach space and let S : X(Γ) → Z be a bounded linear operator. If kSxnk → 0 for every RIS then kSxnk → 0 for every bounded, weakly null sequence, and hence S is compact.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces In a way that is not too different from earlier mixed-Tsirelson constructions, we K show that the space XK = X(Γ ) is HI and that for every operator T on XK there is a scalar λ such that k(T − λI)xnk → 0 for every RIS. So by the crucial lemma we get our main result. Theorem

The space XK has the Scalar-plus-Compact Property.

Constructing XK

To construct XK, we ﬁx in advance a suitable fast-decreasing sequence of weights βi and a fast-increasing sequence of regular families Mi . We then use recursion to build a standard BD-set ΓK with mixed weights, making sure as we add elements of ∆n+1 at stage n + 1 that the history of any element of weight βi is in the family Mi , but also using a coding to restrict the choice of top γ when γ has weight βi with i odd.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces So by the crucial lemma we get our main result. Theorem

The space XK has the Scalar-plus-Compact Property.

Constructing XK

In a way that is not too different from earlier mixed-Tsirelson constructions, we K show that the space XK = X(Γ ) is HI and that for every operator T on XK there is a scalar λ such that k(T − λI)xnk → 0 for every RIS.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Constructing XK

The space XK has the Scalar-plus-Compact Property.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces The scalar-plus-compact property came as a (very agreeable!) surprise.

It seemed natural to ask whether an isomorphic predual of `1 with few operators automatically has very few.Tarbard answered this question in a striking way. Theorem [Tarbard (2012)] For each natural number d there is a well-founded BD-set Γd and a strictly singular operator S on X(Γd ) such that Sd+1 = 0 every bounded linear operator T on X(Γd ) is uniquely expressible as

d X r T = λ0I + λr S + K , r=1

with λ0, . . . , λd scalars and K compact.

Tarbard’s spaces

The original motivation for the construction of the space XK was to ﬁnd a hereditarily indecomposable space with dual isomorphic to `1.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces It seemed natural to ask whether an isomorphic predual of `1 with few operators automatically has very few.Tarbard answered this question in a striking way. Theorem [Tarbard (2012)] For each natural number d there is a well-founded BD-set Γd and a strictly singular operator S on X(Γd ) such that Sd+1 = 0 every bounded linear operator T on X(Γd ) is uniquely expressible as

d X r T = λ0I + λr S + K , r=1

with λ0, . . . , λd scalars and K compact.

Tarbard’s spaces

The original motivation for the construction of the space XK was to ﬁnd a hereditarily indecomposable space with dual isomorphic to `1. The scalar-plus-compact property came as a (very agreeable!) surprise.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Tarbard answered this question in a striking way. Theorem [Tarbard (2012)] For each natural number d there is a well-founded BD-set Γd and a strictly singular operator S on X(Γd ) such that Sd+1 = 0 every bounded linear operator T on X(Γd ) is uniquely expressible as

d X r T = λ0I + λr S + K , r=1

with λ0, . . . , λd scalars and K compact.

Tarbard’s spaces

It seemed natural to ask whether an isomorphic predual of `1 with few operators automatically has very few.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Tarbard’s spaces

d X r T = λ0I + λr S + K , r=1

with λ0, . . . ,Spiros λd scalars A. Argyros and and RichardK G.compact. Haydon Bourgain–Delbaen L∞-spaces Theorem [S.A.–D. Freeman–R.H.–E. Odell–Th. Raikoftsalis–Th. Schlumprecht–D. Zissimopoulou (2012)] Let Y be a separable super-reﬂexive Banach space. Then Y embeds in a space X(Γ) with very few operators.

We note that this result applies in particular when Y is a separable Hilbert space. We conjecture that for spaces Y with separable dual, the only obstruction to embeddability in a space with very few operators is the existence of a subspace of Y isomorphic to c0. While we have some improvements of the above theorem, we cannot prove this “best possible” result.

Embeddings into spaces with very few operators

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces We note that this result applies in particular when Y is a separable Hilbert space. We conjecture that for spaces Y with separable dual, the only obstruction to embeddability in a space with very few operators is the existence of a subspace of Y isomorphic to c0. While we have some improvements of the above theorem, we cannot prove this “best possible” result.

Embeddings into spaces with very few operators

We finish this part of the talk with two results that give partial answers to the question: “For which Banach spaces Y does there exist an isomorphic embedding of Y into a space with very few operators?”. It turns out that the answer is affirmative for some spaces that are in themselves not at all pathological. Theorem [S.A.–D. Freeman–R.H.–E. Odell–Th. Raikoftsalis–Th. Schlumprecht–D. Zissimopoulou (2012)] Let Y be a separable super-reflexive Banach space. Then Y embeds in a space X(Γ) with very few operators.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces We conjecture that for spaces Y with separable dual, the only obstruction to embeddability in a space with very few operators is the existence of a subspace of Y isomorphic to c0. While we have some improvements of the above theorem, we cannot prove this “best possible” result.

Embeddings into spaces with very few operators

We note that this result applies in particular when Y is a separable Hilbert space.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Embeddings into spaces with very few operators

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Let us just say that there is a theory of such things and give the following result as a sample of what is possible.

Theorem [S.A.–R.H.–Th. Raikoftsalis]

There is a standard BD-set Γ (not well-founded!) such that X(Γ) contains `1 and has very few operators.

Spaces with very few operators and non-separable dual

We have not so far said anything about BD-spaces with non-separable dual, or about standard BD-sets that are not well-founded.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Theorem [S.A.–R.H.–Th. Raikoftsalis]

There is a standard BD-set Γ (not well-founded!) such that X(Γ) contains `1 and has very few operators.

Spaces with very few operators and non-separable dual

We have not so far said anything about BD-spaces with non-separable dual, or about standard BD-sets that are not well-founded. Let us just say that there is a theory of such things and give the following result as a sample of what is possible.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Spaces with very few operators and non-separable dual

Theorem [S.A.–R.H.–Th. Raikoftsalis]

There is a standard BD-set Γ (not well-founded!) such that X(Γ) contains `1 and has very few operators.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Plan

L∞-spaces and the general BD-construction Standard BD-spaces The FOS embedding theorem Tsirelson spaces and “few-operators” constructions

The space XK with very few operators The Invariant Subspace Property

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces The Invariant Subspace Problem

Deﬁnition Let X be a Banach space and T a bounded linear operator on X. The operator T admits a non-trivial invariant subspace if there exists a closed subspace Y of X with Y 6= {0}, Y 6= X such that T [Y ] ,→ Y .

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces The Invariant Subspace Problem

Problem Let X be an inﬁnite dimensional separable Banach space and T : X → X a bounded linear operator. Does T admit a non trivial closed invariant subspace?

The above problem remains open in the case of the separable Hilbert space. However there are many results concerning certain classes of operators and Banach spaces.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces The Invariant Subspace Problem

Problem Let X be an inﬁnite dimensional separable Banach space and T : X → X a bounded linear operator. Does T admit a non trivial closed invariant subspace?

The above problem remains open in the case of the separable Hilbert space. However there are many results concerning certain classes of operators and Banach spaces.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces The Invariant Subspace Problem

Problem Let X be an inﬁnite dimensional separable Banach space and T : X → X a bounded linear operator. Does T admit a non trivial closed invariant subspace?

The above problem remains open in the case of the separable Hilbert space. However there are many results concerning certain classes of operators and Banach spaces.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Positive results

In 1954 M. Aronszajn and K. T. Smith proved that every compact operator defined on an infinite dimensional Banach space admits a non trivial closed invariant subspace. In 1973 V. I. Lomonosov generalized the above result, by showing that in a complex Banach space X, every bounded linear operator T 6= λI, for λ ∈ C, which commutes with a non zero compact operator, admits a non trivial closed hyperinvariant subspace. N. D. Hooker (1981) and G. Sirotkin (2005) presented a version of the above theorem for real Banach spaces. In particular, an operator T defined on a real infinite dimensional Banach space, which satisfies the assumptions of Lomonosov’s Theorem, admits a non trivial closed hyperinvariant subspace if and only if (αI − T )2 + β2I 6= 0, for every α, β ∈ R, with β 6= 0.

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Positive results

Spiros A. Argyros and Richard G. Haydon Bourgain–Delbaen L∞-spaces Negative results

P. Enflo and C. J. Read, in papers published in 1987 and 1984 respectively, present the first examples of operators on non reflexive, non classical Banach spaces, not admitting a non trivial closed invariant subspace.

Also, C. J. Read proved that any separable Banach space containing either c0 or complemented `1, admits an operator without a non trivial closed invariant subspace.

Since c0 is separably injective, c0 is a complemented subspace of X. The assumption that `1 is a complemented subspace is necessary by our result with Raikoftsalis, mentioned earlier.

Moreover, in 1999 C. J. Read presents an example of a strictly singular operator on a non reﬂexive Banach space, which does not admit a non trivial closed invariant subspace.