Spaces With Small Operators

d’apresSpiros Argyros & Richard Haydon

A Herdeditarily Indecomposable L∞–spacethat solves the scalar–plus–compact problem

MUIC Seminar

September 21, 2016

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Introduction Prelude

Unusual Banach Spaces I

Theorem (James,1950)Let X be a Banach space with an unconditional basis. Then X isreflexive if and only if neither `1 nor c0 is isomorphic to a subspace ofX .

Theorem (Tsirelson,1974)There is a reflexive and separable Banach space T that contains noisomorphic copy of either c0 or `p, 1 ≤ p <∞.

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Introduction Prelude

Unusual Banach Spaces II

The idea behind the construction of Tsirelson’s is as follows. Take thec00 space i.e. the space of all finite sequences (or the space c00(Γ) –the space of functions with finite support) and construct a norm ‖ · ‖Tso that the completion T = c00

‖·‖T contains neither `p nor c0.The basic tool is to see that the new norm ‖ · ‖ is such that themapping PX0 : T → X0 is not continuous (where X0 is a subspace of Talgebraically isomorphic to, say, `1) . That is, the new norm ‖ · ‖T is soselected that ‖x‖T ≤ C ·

∑γ∈Γ |x(γ)| is not satisfied for any constant C.

Clearly, no such norm can be defined globally. It is through meticulousconstruction of the norm on subspaces and then defining the globalnorm in inductive manner can such norm be shown to exist. The actualconstruction of Tsirelson’s, while simple in concept, is the height ofmathematical virtuosity.

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Introduction Prelude

Entreacte

Professor T showing his students how simple is the construction of theTsirelson Space.

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Introduction Prelude

Entreacte

Professor T showing his students how simple is the construction of theTsirelson Space.

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Introduction Prelude

Definition (Distortable Space)

Let (X , ‖ · ‖) be an infinite–dimensional Banach space and let λ > 1.The space X is said to be λ–distortable if there exists anequivalent norm | · | on X such that, for every infinite–dimensionalsubspace Y of X

sup{|y1||y2|

: ‖y1‖ = ‖y2‖ = 1, y1, y2 ∈ Y}> λ.

X is said to be distortable (resp. arbitrarily distortable ) if X isλ–distortable for some (resp. all) λ > 1.

Theorem (Milman, 1969/71)If X is not distortable then X contains a copy of c0 or `p for some1 ≤ p <∞.

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Introduction Prelude

Corollary

Tsirelson space T is distortable.

In 1991, T. Schlumprecht produced the famous example of anarbitrarily distortable ”Tsirelson – like” Banach space S. Therefinement of construction of this type of Banach space led to theGowers – Maurey construction of the Banach space with nounconditional basic sequence

Theorem (Schlumprecht, 1991)

There exists a Banach space S (the Schlumprecht space) which isarbitrarily distortable.

Theorem (Gowers, 1994)There exists a Banach space Xgm not containing c0, `1 or a reflexivesubspace.

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Introduction Problem

An Old Problem

ProblemDoes there exist a Banach space X such that every T ∈ L(X ,X ) is ofthe form T = λI + K , where λ ∈ R and K ∈ K (X ,X ) is a compactoperator.

This problem was first stated explicitely by J. Lindenstrauss in 1976 butits roots are in the early paper of N. Aronszajn and K. T. Smith in 1954.

The discovery of a pathological Banach space Xg,m by T. Gowers andB. Maurey in 1993 and subsequent work by T. Gowers along this linesshed a new light onto the problem.

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Introduction HI Space

Strictly Singular Operators

Definition

An operator T : X → X is called strictly singular if for every closedsubspace Y of X such that T|Y is an isomorphism, we havedim Y < +∞.

Definition

A Banach space X is indecomposable if there do not exist infinitedimensional closed subspaces Y and Z of X with X = Y ⊕ Z ;

A Banach space X is Hereditarily Indecomposable (HI space) ifevery closed subspace of X is indecomposable.

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Introduction HI Space

Strictly Singular Operators

Definition

An operator T : X → X is called strictly singular if for every closedsubspace Y of X such that T|Y is an isomorphism, we havedim Y < +∞.

Definition

A Banach space X is indecomposable if there do not exist infinitedimensional closed subspaces Y and Z of X with X = Y ⊕ Z ;A Banach space X is Hereditarily Indecomposable (HI space) ifevery closed subspace of X is indecomposable.

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Introduction HI Space

Theorem (Gowers–Maurey, 1993)

There exists an HI space.

NoteThe HI space of the Theorem is usually denoted Xgm.

Theorem (Gowers (1999), Gasparis (2003))

Every operator T : Xgm → Xgm is of the form T = λI + S, where S is astrictly singular operator.

TheoremLet X and Y be Banach spaces. Every compact operator T ∈ L(X ,Y )is strictly singular but there exist operators which are strictly singularbut not compact.

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Main Theorem Strictly Singular Operators

TheoremAn operator between Hilbert spaces is compact if and only if it isstrictly singular.

Example

If 1 ≤ p < r <∞, then the natural embedding J : `p → `r is anon–compact strictly singular operator.

Every operator T on the HI space Xgm is a strictly singular perturbationof the identity but not a compact perturbation of the identity (i.e. is nota solution to the question of Lindenstrauss), as was shown byG. Androulakis and Th. Schlumprecht in 2001.

Theorem (Androulakis–Schlumprecht, 2001)

There exists a strictly singular, non–compact operator on the spaceXgm.

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Main Theorem Fredholm Operators

Basic properties of Fredholm Operators

Definition

An operator T ∈ L(X ,Y ) is a Fredholm Operator if T (X ) is closedand ker T and coker T = Y/T (X ) are finite–dimensional.n(T ) :=dim ker T , d(T ) := dim coker T .index of T i(T ) := n(T )− d(T )

TheoremA bounded linear operator is Fredholm if and only if both n(T ) andd(T ) are finite;A bounded linear operator T : X → Y between Banach spaces isFredholm if and only if the operator TC : XC → YC is Fredholm. Inthis case i(TC) = i(T );If T is a Fredholm operator with index i and S is strictly singularthen T + S is Fredholm with index i.

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Main Theorem Some Properties of the Xgm space

Properties of the Xgm space

(A) A non–trivial projection P (i.e. PX and (I − P)X are bothinfinite–dimensional) is not of the form λI + S;

(B) If T is Fredholm with index i and S is strictly singular then T + S isFredholm with index i ;

(C) From (A) + (B)⇒ X 6= Y ⊕ Z with both Y and Zinfinite–dimensional ;

(D) X 6∼= Y for any proper infinite–dimensional subspace Y of X , forany isomorphism X → Y is Fredholm with a non–zero index, whileλI + S has zero index .

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Main Theorem L∞ spaces

Definition

A separable Banach space X is an L∞,λ–space if there is andincreasing sequence (Fn)n∈N of finite dimensional subspaces of Xsuch that the union

⋃n∈N Fn is dense in X and, for each n, Fn is

λ–isomorphic to `dim Fn∞ :

λ–isomorphic⇔ ∃ isomorphism µ : Fn → `dim Fn∞ with

‖µ‖ · ‖µ−1‖ < λ;We say that X is an L∞–space if it is an L∞,λ–space for someλ > 1.

Theorem (Lewis–Stegall, 1971)

If a separable L∞–space X has no subspace isomorphic to `1 then itsdual X ∗is isomorphic to `1.

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Main Theorem L∞ spaces

This implies that the dual of a separable HI L∞–space isisomorphic to `1;

Bourgain and Delbaen (1980) constructed an L∞–space Xa,bwhich does not contain c0.

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Main Theorem Theorem of Argyros and Haydon

Theorem (Spiros A. Argyros and Richard G. Haydon)

There exists a hereditarily indecomposable L∞–space XK with dualisomorphic to `1 such that every bounded linear operator T on thisspace is of the form T = λI + K , where K ∈ K (X ).

Proof.

The space XK of the theorem is a modification of Xgm with specificdesign to be an L∞–space so that its dual is isomorphic to `1. The factthat it has only compact perturbations of identity as operators was,according to the authors, a ”lucky shot”.

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Main Theorem Invariant Subspaces

Definition (Invariant Subspace)

Let T : X → X be a bounded linear operator on a Banach space X andlet V be a proper subspace of X (i.e. ∅ 6= V 6= X ). We say that V isinvariant under T if T (V ) ⊆ V .

ProblemWhen does a bounded linear operator on an infinite dimensionalBanach space have a non–trivial closed invariant subspace ?

P. Enflo (1987) showed an example of a bounded linear operatoron a real or complex separable non–reflexive space Banachspace without non–trivial invariant subspaces;C. J. Read (1986/87) showed an example of a bounded linearoperator on the space `1 without non–trivial closed invariantsubspaces.

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Main Theorem Consequences of the Main Theorem

Lomonosov Theorem I

Theorem (Lomonosov Theorem)If a bounded linear operator T on an infinite dimensional real orcomplex Banach space commutes with a non–zero compact operatorthen T has a non–trivial closed invariant subspace.

Corollary

Every operator T on the space XK of Argyros and Haydon has anon–trivial invariant subspace.

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Main Theorem Consequences of the Main Theorem

Lomonosov Theorem II

Proof.

TK = (λI + K )K = λK + K 2 = K (λI + K ) = KT

for every K ∈ K (X ), K 6= 0. By the Lomonosov Theorem T has anon–trivial invariant subspace. If K = 0 then T trivially commutes withany non–zero compact operator.

NoteThe space XK is the first (and so far only known) infinite–dimensionalBanach space such that every operator on it has a non–trivial invariantsubspace.

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Open Problems General Operators on Banach Spaces

Finite Rank Operators

Definition (Finite Rank Operator)

Let X ,Y be Banach spaces and let T ∈ L(X ,Y ). The mapping Tis called a finite rank operator if its image space is finitedimensional.Each finite rank mapping T can be represented in the form

Tx =n∑

i=1

〈x , x∗i 〉yi

for x ∈ X with yi ∈ Y and xi ∈ X for i = 1, . . . ,n.The collection of all finite rank operators from X to Y is a linearsubspace of L(X ,Y ) and is denoted by A (X ,Y ).

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Open Problems Nuclear Operators

Nuclear Operators I

Definition (Nuclear Operator)

Let X ,Y be normed spaces and let U,V be closed unit balls in X resp.Y . Let pV and pU∗ be Minkowski functionals of V resp. U∗. Anoperator T ∈ L(X ,Y ) is a nuclear operator if there exist sequencesx∗n ∈ X ∗ and yn ∈ Y , (n ∈ N) with∑

n∈Npu∗(x∗n )pv (yn) <∞

such thatTx =

∑n∈N〈x , x∗n 〉yn for x ∈ X .

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Open Problems Nuclear Operators

Nuclear Operators II

Definition

For each nuclear mapping T : X → Y we set

ν(T ) := inf

{∑n∈N

pu∗(x∗n )pv (yn)

}

where the infimum is taken over all possible representations of T .The function ν is a norm on N (X ,Y ).

TheoremFor a normed space X and a Banach space Y the space (N (X ,Y ), ν)is a Banach space.

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Open Problems Nuclear Operators

Nuclear Operators III

TheoremThe space of all finite rank operators A (X ,Y ) is dense in the space ofall nuclear operators N (X ,Y ).

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Open Problems Nuclear Operators

Natural Operators on Banach Spaces

Natural Operators

The following types of operators T : X → X exist on all Banach spacesX :

(Multiple of) identity λI;Finite rank operators A (X );Nuclear operators N (X ) .

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Open Problems Open Problems

Problem (1)Does every operator on a Hilbert space have a non–trivial invariantsubspace ?

Problem (2)Does there exist an infinite–dimensional Banach X space such thatevery operator T ∈ L(X ) is of the form T = λI + N, where N ∈ N (X ) ?

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Open Problems Open Problems

G. Androulakis and T. Schlumprecht, Strictly singular,non-compact operators exist on the space of Gowers and Maurey,J. London Math. Soc. (2) 64 (2001), no. 3, 655-674.

N. Aronszajn and K. T. Smith, Invariant subspaces of completelycontinuous operators, Annals of Math. (2) 60 (1954), 345–350 .

J. Bourgain and F. Delbaen, A class of special L∞ spaces, ActaMath. 145 (1980), no. 3–4, 155-176.

Per Enflo, On the invariant subspace problem for Banach spaces,Acta Math. 158 (1987), no. 3–4, 213-313.

I. Gasparis, Strictly singular non–compact operators onhereditarily indecomposable Banach spaces, Proc. Amer. Math.Soc. 131 (2003), 1181–1189 .

W.T. Gowers, A Banach space not containing c0, `1 or a reflexivesubspace, Trans. Amer. Math. Soc. 344 (1994), no. 1, 407-420.

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Open Problems Open Problems

W.T. Gowers, A remark about the scalar–plus–compact problem, inConvex Geometric Analysis (Berkeley, CA, 1996), Math. Sci. Res.Inst. Publ. 34, Cambridge Univ. Press, Cambridge, 1999

W. T. Gowers and B. Maurey, The unconditional basic sequenceproblem, J. Amer. Math. Soc. 6 (1993), 851–874 .

Robert C. James, Bases and reflexivity of Banach spaces, Ann. ofMath. (2) 52 (1950), 518–527.

D. R. Lewis and C. Stegall, Banach spaces whose duals areisomorphic to `1(Γ), J. Functional Analysis 12 (1973), 177-187.

J. Lindenstrauss, Some open problems in Banach space theory,Seminaire Choquet, Initiation a l’analyse, 15 (1975–76), Expose18, 9 p.

Vitali D. Milman, Geometric theory of Banach spaces. II. Geometryof the unit ball. (Russian), Uspehi Mat. Nauk 26 (1971), no. 6(162),73-149.

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Open Problems Open Problems

– James classes of minimal systems, and their connection with theisometry properties of B-spaces, (Russian) Dokl. Akad. NaukSSSR 192 1970, 742-745.

C. J. Read, A solution to the invariant subspace problem, Bull.London Math. Soc. 16 (1984), no. 4, 337-401.–, A solution to the invariant subspace problem on the space `1,Bull. London Math. Soc. 17 (1985), no. 4, 305-317.

Thomas Schlumprecht, An arbitrarily distortable Banach space,Israel J. Math. 76 (1991), no. 1–2, 81-95.

Boris S. Tsirelson, Not every Banach space contains animbedding of `p or c0, (Russian) Functional Anal. Appl. 8(1974),138-141.

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