Banach spaces with rich L8

Spiros A. Argyros

Department of Mathematics National Technical University of Athens

Athens, Greece

First Brazilian Workshop in Geometry of Banach spaces Maresias, August 2014

A separable spaceX is aL8space, if there exists a constantC¡0 and an increasing sequence of finite dimensional spacespFnqnsuch that eachFnis C-isomorphic to`₈pdimF_nqandYnF_nX.

(Lewis - Stegall)IfX is a separableL8space thenX`₁ orX Mr0,1s.

(Pelczynski)IfXis non-separable thenX isomorphically contains`₁.

A separable spaceX is aL8space, if there exists a constantC¡0 and an increasing sequence of finite dimensional spacespFnqnsuch that eachFnis C-isomorphic to`₈pdimF_nqandYnF_nX.

(Lewis - Stegall)IfX is a separableL8space thenX`₁ orX Mr0,1s.

(Pelczynski)IfXis non-separable thenX isomorphically contains`₁.

A separable spaceX is aL8space, if there exists a constantC¡0 and an increasing sequence of finite dimensional spacespFnqnsuch that eachFnis C-isomorphic to`₈pdimF_nqandYnF_nX.

(Lewis - Stegall)IfX is a separableL8space thenX`₁ orX Mr0,1s.

(Pelczynski)IfXis non-separable thenX isomorphically contains`₁.

A separable spaceX is aL8space, if there exists a constantC¡0 and an increasing sequence of finite dimensional spacespFnqnsuch that eachFnis C-isomorphic to`₈pdimF_nqandYnF_nX.

(Lewis - Stegall)IfX is a separableL8space thenX`₁ orX Mr0,1s.

(Pelczynski)IfXis non-separable thenX isomorphically contains`₁.

In 1980J. Bourgain and F. Delbaenconstructed the first L8space not containingc₀.

Bourgain-Delbaen method was a critical ingredient for the solution of the "scalar-plus-compact" problem.

In 1980J. Bourgain and F. Delbaenconstructed the first L8space not containingc₀.

Bourgain-Delbaen method was a critical ingredient for the solution of the "scalar-plus-compact" problem.

In 1980J. Bourgain and F. Delbaenconstructed the first L8space not containingc₀.

Bourgain-Delbaen method was a critical ingredient for the solution of the "scalar-plus-compact" problem.

Theorem (S. A., R. Haydon 2011)

There exists aL8Hereditarily Indecomposable Banach space X_K such thatX_K is isomorphic to`1pNqand everyT :X_K ÑX_K is of the formT λI K withK a compact operator.

This is the first example of a Banach spaceX such that every operatorT PLpXqadmits a non trivial closed invariant subspace.

Theorem (S. A., R. Haydon 2011)

There exists aL8Hereditarily Indecomposable Banach space X_K such thatX_K is isomorphic to`1pNqand everyT :X_K ÑX_K is of the formT λI K withK a compact operator.

This is the first example of a Banach spaceX such that every operatorT PLpXqadmits a non trivial closed invariant subspace.

Problem

Does there exist a separable Banach spaceXwith the hereditary "scalar-plus-compact" property? i.e. for everyY closed subspace ofXevery operatorT PLpY,Xqis of the form T λIY,X K withK PKpY,Xq.

Problem (weaker version)

Does there existsXsaturated by infinite dimensional closed subspaces each one satisfying the "scalar-plus-compact"

property?

Problem

Does there exist a separable Banach spaceXwith the hereditary "scalar-plus-compact" property? i.e. for everyY closed subspace ofXevery operatorT PLpY,Xqis of the form T λIY,X K withK PKpY,Xq.

Problem (weaker version)

Does there existsXsaturated by infinite dimensional closed subspaces each one satisfying the "scalar-plus-compact"

property?

Problem

Does there exist a separable Banach spaceXwith the hereditary "scalar-plus-compact" property? i.e. for everyY closed subspace ofXevery operatorT PLpY,Xqis of the form T λIY,X K withK PKpY,Xq.

Problem (weaker version)

Does there existsXsaturated by infinite dimensional closed subspaces each one satisfying the "scalar-plus-compact"

property?

Problem (Ramsey Property for "s c")

LetXbe a separable Banach space. Does there exist an infinite dimensional closed subspaceY ofXsuch that either:

Every further subspace ofY satisfies the "scalar-plus-compact"

property.

or

Every infinite dimensional closed subspace ofY does not satisfy the "scalar-plus-compact" property.

Problem (weaker version)

Assuming that the spaceXhas a Schauder basis, what is the answer to the above for the class of all block subspaces ofX.

Problem (Ramsey Property for "s c")

LetXbe a separable Banach space. Does there exist an infinite dimensional closed subspaceY ofXsuch that either:

Every further subspace ofY satisfies the "scalar-plus-compact"

property.

or

Every infinite dimensional closed subspace ofY does not satisfy the "scalar-plus-compact" property.

Problem (weaker version)

Assuming that the spaceXhas a Schauder basis, what is the answer to the above for the class of all block subspaces ofX.

Problem (Ramsey Property for "s c")

LetXbe a separable Banach space. Does there exist an infinite dimensional closed subspaceY ofXsuch that either:

Every further subspace ofY satisfies the "scalar-plus-compact"

property.

or

Every infinite dimensional closed subspace ofY does not satisfy the "scalar-plus-compact" property.

Problem (weaker version)

Assuming that the spaceXhas a Schauder basis, what is the answer to the above for the class of all block subspaces ofX.

Problem (Ramsey Property for "s c")

LetXbe a separable Banach space. Does there exist an infinite dimensional closed subspaceY ofXsuch that either:

Every further subspace ofY satisfies the "scalar-plus-compact"

property.

or

Every infinite dimensional closed subspace ofY does not satisfy the "scalar-plus-compact" property.

Problem (weaker version)

Assuming that the spaceXhas a Schauder basis, what is the answer to the above for the class of all block subspaces ofX.

Problem (Ramsey Property for "s c")

LetXbe a separable Banach space. Does there exist an infinite dimensional closed subspaceY ofXsuch that either:

Every further subspace ofY satisfies the "scalar-plus-compact"

property.

or

Every infinite dimensional closed subspace ofY does not satisfy the "scalar-plus-compact" property.

Problem (weaker version)

Assuming that the spaceXhas a Schauder basis, what is the answer to the above for the class of all block subspaces ofX.

Problem (Ramsey Property for "s c")

LetXbe a separable Banach space. Does there exist an infinite dimensional closed subspaceY ofXsuch that either:

Every further subspace ofY satisfies the "scalar-plus-compact"

property.

or

Every infinite dimensional closed subspace ofY does not satisfy the "scalar-plus-compact" property.

Problem (weaker version)

Assuming that the spaceXhas a Schauder basis, what is the answer to the above for the class of all block subspaces ofX.

Problem

Does there exists areflexiveBanach spaceXwith the

"scalar-plus-compact" property?

Problem

Does there exists areflexiveBanach spaceXwith the

"scalar-plus-compact" property?

Theorem (S.A, Pavlos Motakis, Proc.LMS (2014))

There exists a reflexive HI Banach spaceX_ISP satisfying the following:

(i) For everyY closed subspace ofX_ISP, everyT PLpYqis of the formT λI S, withSstrictly singular.

(ii) For everyY andQ,S,T :Y ÑY strictly singular operators,the compositionQST is a compact one.

Every Banach space satisfying (i) and (ii), also satisfies the hereditary Invariant Subspace Property(i.e. for every infinite dimensional closed subspaceY ofX_ISP and every operatorT PLpYqadmits a non trivial closed invariant subspace).

Theorem (S.A, Pavlos Motakis, Proc.LMS (2014))

There exists a reflexive HI Banach spaceX_ISP satisfying the following:

(i) For everyY closed subspace ofX_ISP, everyT PLpYqis of the formT λI S, withSstrictly singular.

(ii) For everyY andQ,S,T :Y ÑY strictly singular operators,the compositionQST is a compact one.

Every Banach space satisfying (i) and (ii), also satisfies the hereditary Invariant Subspace Property(i.e. for every infinite dimensional closed subspaceY ofX_ISP and every operatorT PLpYqadmits a non trivial closed invariant subspace).

Theorem (S.A, Pavlos Motakis, Proc.LMS (2014))

There exists a reflexive HI Banach spaceX_ISP satisfying the following:

(i) For everyY closed subspace ofX_ISP, everyT PLpYqis of the formT λI S, withSstrictly singular.

(ii) For everyY andQ,S,T :Y ÑY strictly singular operators,the compositionQST is a compact one.

Every Banach space satisfying (i) and (ii), also satisfies the hereditary Invariant Subspace Property(i.e. for every infinite dimensional closed subspaceY ofX_ISP and every operatorT PLpYqadmits a non trivial closed invariant subspace).

Theorem (S.A, Pavlos Motakis, Proc.LMS (2014))

There exists a reflexive HI Banach spaceX_ISP satisfying the following:

(i) For everyY closed subspace ofX_ISP, everyT PLpYqis of the formT λI S, withSstrictly singular.

(ii) For everyY andQ,S,T :Y ÑY strictly singular operators,the compositionQST is a compact one.

Every Banach space satisfying (i) and (ii), also satisfies the hereditary Invariant Subspace Property(i.e. for every infinite dimensional closed subspaceY ofX_ISP and every operatorT PLpYqadmits a non trivial closed invariant subspace).

Theorem (S.A, Pavlos Motakis, Proc.LMS (2014))

There exists a reflexive HI Banach spaceX_ISP satisfying the following:

(i) For everyY closed subspace ofX_ISP, everyT PLpYqis of the formT λI S, withSstrictly singular.

(ii) For everyY andQ,S,T :Y ÑY strictly singular operators,the compositionQST is a compact one.

Every Banach space satisfying (i) and (ii), also satisfies the hereditary Invariant Subspace Property(i.e. for every infinite dimensional closed subspaceY ofX_ISP and every operatorT PLpYqadmits a non trivial closed invariant subspace).

A Banach spaceX is said to beL8saturatedif every closed infinite dimension subspaceY ofX contains a further subspace which is anL8 space.

The known examples ofL8 saturated Banach spaces are the classical ones, namely the spacec₀and the spaces Cpαqwhereαis an infinite ordinal number with the usual order topology.

A Banach spaceX is said to beL8saturatedif every closed infinite dimension subspaceY ofX contains a further subspace which is anL8 space.

The known examples ofL8 saturated Banach spaces are the classical ones, namely the spacec₀and the spaces Cpαqwhereαis an infinite ordinal number with the usual order topology.

A Banach spaceX is said to beL8saturatedif every closed infinite dimension subspaceY ofX contains a further subspace which is anL8 space.

The known examples ofL8 saturated Banach spaces are the classical ones, namely the spacec₀and the spaces Cpαqwhereαis an infinite ordinal number with the usual order topology.

Problem

Does there exist a Banach spaceX which isL8 saturated and does not contain a subspace isomorphic toc₀?

The problem was stated by H.P. Rosenthal in the 80’s and in particular he was interested for a negative answer.

If such a space exists then passing to a subspace we may assume that aL8space exists which isL8 saturated and does not containc₀.

AnyL8space which isL8saturated has separable dual.

Moreover, if it does not containc₀then it does not contain unconditional basic sequences. Hence, byGowers Dichotomyit contains an HI subspace.

Problem

Does there exist a Banach spaceX which isL8 saturated and does not contain a subspace isomorphic toc₀?

The problem was stated by H.P. Rosenthal in the 80’s and in particular he was interested for a negative answer.

If such a space exists then passing to a subspace we may assume that aL8space exists which isL8 saturated and does not containc₀.

AnyL8space which isL8saturated has separable dual.

Moreover, if it does not containc₀then it does not contain unconditional basic sequences. Hence, byGowers Dichotomyit contains an HI subspace.

Problem

Does there exist a Banach spaceX which isL8 saturated and does not contain a subspace isomorphic toc₀?

The problem was stated by H.P. Rosenthal in the 80’s and in particular he was interested for a negative answer.

If such a space exists then passing to a subspace we may assume that aL8space exists which isL8 saturated and does not containc₀.

AnyL8space which isL8saturated has separable dual.

Moreover, if it does not containc₀then it does not contain unconditional basic sequences. Hence, byGowers Dichotomyit contains an HI subspace.

Problem

Does there exist a Banach spaceX which isL8 saturated and does not contain a subspace isomorphic toc₀?

The problem was stated by H.P. Rosenthal in the 80’s and in particular he was interested for a negative answer.

If such a space exists then passing to a subspace we may assume that aL8space exists which isL8 saturated and does not containc₀.

AnyL8space which isL8saturated has separable dual.

Moreover, if it does not containc₀then it does not contain unconditional basic sequences. Hence, byGowers Dichotomyit contains an HI subspace.

Problem

Does there exist a Banach spaceX which isL8 saturated and does not contain a subspace isomorphic toc₀?

The problem was stated by H.P. Rosenthal in the 80’s and in particular he was interested for a negative answer.

If such a space exists then passing to a subspace we may assume that aL8space exists which isL8 saturated and does not containc₀.

AnyL8space which isL8saturated has separable dual.

Moreover, if it does not containc₀then it does not contain unconditional basic sequences. Hence, byGowers Dichotomyit contains an HI subspace.

A solution to the Rosenthal’s Problem will yield answers to some of the problems stated before.

The aim of the present talk is to present some recent joint work withPavlos Motakisrelated to Rosenthal’s problem.

Theorem

There exists aL8spaceXwith a basispd_nqnsuch that for every infinite subsetLofNthe subspaceX_L  d_n:nPL¡ contains aL8space.

A solution to the Rosenthal’s Problem will yield answers to some of the problems stated before.

The aim of the present talk is to present some recent joint work withPavlos Motakisrelated to Rosenthal’s problem.

Theorem

There exists aL8spaceXwith a basispd_nqnsuch that for every infinite subsetLofNthe subspaceX_L  d_n:nPL¡ contains aL8space.

A solution to the Rosenthal’s Problem will yield answers to some of the problems stated before.

The aim of the present talk is to present some recent joint work withPavlos Motakisrelated to Rosenthal’s problem.

Theorem

There exists aL8spaceXwith a basispd_nqnsuch that for every infinite subsetLofNthe subspaceX_L  d_n:nPL¡ contains aL8space.

A solution to the Rosenthal’s Problem will yield answers to some of the problems stated before.

The aim of the present talk is to present some recent joint work withPavlos Motakisrelated to Rosenthal’s problem.

Theorem

There exists aL8spaceXwith a basispd_nqnsuch that for every infinite subsetLofNthe subspaceX_L  d_n:nPL¡ contains aL8space.

A BDL8 space is a separable subspaceXof`₈pΓqwith the supremum norm.

The setΓis countable,Γ Y⁸_n1ΓnwherepΓnqis an increasing sequence of finite sets.

We set∆₁Γ₁and∆_{n 1}Γ_{n 1}zΓ_n. There existsC ¡0 and extension operators

in:`<sub>8</sub>pΓnq Ñ`₈pΓq

(i.e. i_npxq|Γn x) such that}i_n} ¤Cfor everynPN. HenceinisC- isomorphic embedding for everynPN.

A BDL8 space is a separable subspaceXof`₈pΓqwith the supremum norm.

The setΓis countable,Γ Y⁸_n1ΓnwherepΓnqis an increasing sequence of finite sets.

We set∆₁Γ₁and∆_{n 1}Γ_{n 1}zΓ_n. There existsC ¡0 and extension operators

in:`<sub>8</sub>pΓnq Ñ`₈pΓq

(i.e. i_npxq|Γn x) such that}i_n} ¤Cfor everynPN. HenceinisC- isomorphic embedding for everynPN.

A BDL8 space is a separable subspaceXof`₈pΓqwith the supremum norm.

The setΓis countable,Γ Y⁸_n1ΓnwherepΓnqis an increasing sequence of finite sets.

We set∆₁Γ₁and∆_{n 1}Γ_{n 1}zΓ_n. There existsC ¡0 and extension operators

in:`<sub>8</sub>pΓnq Ñ`₈pΓq

(i.e. i_npxq|Γn x) such that}i_n} ¤Cfor everynPN. HenceinisC- isomorphic embedding for everynPN.

A BDL8 space is a separable subspaceXof`₈pΓqwith the supremum norm.

The setΓis countable,Γ Y⁸_n1ΓnwherepΓnqis an increasing sequence of finite sets.

We set∆₁Γ₁and∆_{n 1}Γ_{n 1}zΓ_n. There existsC ¡0 and extension operators

in:`<sub>8</sub>pΓnq Ñ`₈pΓq

(i.e. i_npxq|Γn x) such that}i_n} ¤Cfor everynPN. HenceinisC- isomorphic embedding for everynPN.

A BDL8 space is a separable subspaceXof`₈pΓqwith the supremum norm.

The setΓis countable,Γ Y⁸_n1ΓnwherepΓnqis an increasing sequence of finite sets.

We set∆₁Γ₁and∆_{n 1}Γ_{n 1}zΓ_n. There existsC ¡0 and extension operators

in:`<sub>8</sub>pΓnq Ñ`₈pΓq

(i.e. i_npxq|Γn x) such that}i_n} ¤Cfor everynPN. HenceinisC- isomorphic embedding for everynPN.

A BDL8 space is a separable subspaceXof`₈pΓqwith the supremum norm.

The setΓis countable,Γ Y⁸_n1ΓnwherepΓnqis an increasing sequence of finite sets.

We set∆₁Γ₁and∆_{n 1}Γ_{n 1}zΓ_n. There existsC ¡0 and extension operators

in:`<sub>8</sub>pΓnq Ñ`₈pΓq

(i.e. i_npxq|Γn x) such that}i_n} ¤Cfor everynPN. HenceinisC- isomorphic embedding for everynPN.

The operatorsi_nare compatible. Namely, forn k

`<sup>∞</sup>(Γn) `^∞(Γk)

`^∞(Γ) PΓk◦in

ik

in

Forγ P∆qwe setd_γi_qpe_γqand X xtdγ: γ PΓuy

The operatorsi_nare compatible. Namely, forn k

`<sup>∞</sup>(Γn) `^∞(Γk)

`^∞(Γ) PΓk◦in

ik

in

Forγ P∆qwe setd_γi_qpe_γqand X xtdγ: γ PΓuy

The operatorsi_nare compatible. Namely, forn k

`<sup>∞</sup>(Γn) `^∞(Γk)

`^∞(Γ) PΓk◦in

ik

in

Forγ P∆qwe setd_γi_qpe_γqand X xtdγ: γ PΓuy

The operatorsi_nare compatible. Namely, forn k

`<sup>∞</sup>(Γn) `^∞(Γk)

`^∞(Γ) PΓk◦in

ik

in

Forγ P∆qwe setd_γi_qpe_γqand X xtdγ: γ PΓuy

The operatorsi_nare compatible. Namely, forn k

`<sup>∞</sup>(Γn) `^∞(Γk)

`^∞(Γ) PΓk◦in

ik

in

Forγ P∆qwe setd_γi_qpe_γqand X xtdγ: γ PΓuy

For everyγ P∆_{q 1}we define a linear functional c_γ e_γiq :`₈pΓqq ÑR.

For everyp¤q andx P`₈pΓpq

i_ppxqpγq c_γpi_qpi_ppxq|Γqqq

The above explains that the construction of specific BD spaces essentially concerns the definition of the functionalsc_γ, γPΓwhich by induction determine the extension operatorsi_q,qPN.

For everyγ P∆_{q 1}we define a linear functional c_γ e_γiq :`₈pΓqq ÑR.

For everyp¤q andx P`₈pΓpq

i_ppxqpγq c_γpi_qpi_ppxq|Γqqq

The above explains that the construction of specific BD spaces essentially concerns the definition of the functionalsc_γ, γPΓwhich by induction determine the extension operatorsi_q,qPN.

For everyγ P∆_{q 1}we define a linear functional c_γ e_γiq :`₈pΓqq ÑR.

For everyp¤q andx P`₈pΓpq

i_ppxqpγq c_γpi_qpi_ppxq|Γqqq

The above explains that the construction of specific BD spaces essentially concerns the definition of the functionalsc_γ, γPΓwhich by induction determine the extension operatorsi_q,qPN.

For everyγ P∆_{q 1}we define a linear functional c_γ e_γiq :`₈pΓqq ÑR.

For everyp¤q andx P`₈pΓpq

i_ppxqpγq c_γpi_qpi_ppxq|Γqqq

The above explains that the construction of specific BD spaces essentially concerns the definition of the functionalsc_γ, γPΓwhich by induction determine the extension operatorsi_q,qPN.

The spaceXis aCL8space.

Indeed, forqPNxtd_γ|Γq : γ PΓquy `₈pΓqq.

SettingF_qi_qr`₈p∆_qqs,pF_qqq is anFDDfor the spaceX.

In particular,td_γ :γ PΓuis a Schauder basis for the space X.

ForIan interval ofNwe denote withP_I the projection with respect to the FDDpF_qqq.

The spaceXis aCL8space.

Indeed, forqPNxtd_γ|Γq : γ PΓquy `₈pΓqq.

SettingF_qi_qr`₈p∆_qqs,pF_qqq is anFDDfor the spaceX.

In particular,td_γ :γ PΓuis a Schauder basis for the space X.

ForIan interval ofNwe denote withP_I the projection with respect to the FDDpF_qqq.

The spaceXis aCL8space.

Indeed, forqPNxtd_γ|Γq : γ PΓquy `₈pΓqq.

SettingF_qi_qr`₈p∆_qqs,pF_qqq is anFDDfor the spaceX.

In particular,td_γ :γ PΓuis a Schauder basis for the space X.

ForIan interval ofNwe denote withP_I the projection with respect to the FDDpF_qqq.

The spaceXis aCL8space.

Indeed, forqPNxtd_γ|Γq : γ PΓquy `₈pΓqq.

SettingF_qi_qr`₈p∆_qqs,pF_qqq is anFDDfor the spaceX.

In particular,td_γ :γ PΓuis a Schauder basis for the space X.

ForIan interval ofNwe denote withP_I the projection with respect to the FDDpF_qqq.

The spaceXis aCL8space.

Indeed, forqPNxtd_γ|Γq : γ PΓquy `₈pΓqq.

SettingF_qi_qr`₈p∆_qqs,pF_qqq is anFDDfor the spaceX.

In particular,td_γ :γ PΓuis a Schauder basis for the space X.

ForIan interval ofNwe denote withP_I the projection with respect to the FDDpF_qqq.

We denote withe_γthe evaluation functional restricted on X. Then the familyte_γ: γ PΓuis equivalent to`1basis.

Moreover iftd_γ : γPΓuis shrinking, thente_γ: γ PΓu generates the dual ofX.

The biorthogonalstd_γ :γ PΓuare defined as d_γ e_γc_γ.

We denote withe_γthe evaluation functional restricted on X. Then the familyte_γ: γ PΓuis equivalent to`1basis.

Moreover iftd_γ : γPΓuis shrinking, thente_γ: γ PΓu generates the dual ofX.

The biorthogonalstd_γ :γ PΓuare defined as d_γ e_γc_γ.

We denote withe_γthe evaluation functional restricted on X. Then the familyte_γ: γ PΓuis equivalent to`1basis.

Moreover iftd_γ : γPΓuis shrinking, thente_γ: γ PΓu generates the dual ofX.

The biorthogonalstd_γ :γ PΓuare defined as d_γ e_γc_γ.

Forγ PΓ_{n 1}zΓ_n,e_γd_γ c_γ where c_γ:`₈pΓnq ÑR

d_γ|inp`₈pΓnqq0

The critical point of the BD constructions is the definition of the functionalsc_γ. The variety of the method used in their definition yields spaces with divergent properties.

Forγ PΓ_{n 1}zΓ_n,e_γd_γ c_γ where c_γ:`₈pΓnq ÑR

d_γ|inp`₈pΓnqq0

The critical point of the BD constructions is the definition of the functionalsc_γ. The variety of the method used in their definition yields spaces with divergent properties.

Forγ PΓ_{n 1}zΓ_n,e_γd_γ c_γ where c_γ:`₈pΓnq ÑR

d_γ|inp`₈pΓnqq0

The critical point of the BD constructions is the definition of the functionalsc_γ. The variety of the method used in their definition yields spaces with divergent properties.

Forγ PΓ_{n 1}zΓ_n,e_γd_γ c_γ where c_γ:`₈pΓnq ÑR

d_γ|inp`₈pΓnqq0

The critical point of the BD constructions is the definition of the functionalsc_γ. The variety of the method used in their definition yields spaces with divergent properties.

Forγ PΓ_{n 1}zΓ_n,e_γd_γ c_γ where c_γ:`₈pΓnq ÑR

d_γ|inp`₈pΓnqq0

The critical point of the BD constructions is the definition of the functionalsc_γ. The variety of the method used in their definition yields spaces with divergent properties.

We define the functionalstc_γ : γ PΓuin a similar manner as the functionals in the norming set of a Mixed Tsirelson spaces.

We start with two strictly increasing sequences of natural numberspm_jqj,pn_jqj which satisfy certain growth

conditions. Among others, _m^nj

j Ñ 8andm₁¥8.

We define the functionalstc_γ : γ PΓuin a similar manner as the functionals in the norming set of a Mixed Tsirelson spaces.

We start with two strictly increasing sequences of natural numberspm_jqj,pn_jqj which satisfy certain growth

conditions. Among others, _m^nj

j Ñ 8andm₁¥8.

We define the functionalstc_γ : γ PΓuin a similar manner as the functionals in the norming set of a Mixed Tsirelson spaces.

We start with two strictly increasing sequences of natural numberspm_jqj,pn_jqj which satisfy certain growth

conditions. Among others, _m^nj

j Ñ 8andm₁¥8.

There are three types of functionals.

LetγP∆_{q 1}. (1)

c_γ0 Thenwpγq 0, apγq 0.

(2)

c_γ 1

m_je_ξP_I Thenwpγq m_j, apγq 1.

(3)

c_γ e_η 1

m_je_ξP_I Thenwpγq m_j, apγq apηq 1¤n_j.

There are three types of functionals.

LetγP∆_{q 1}. (1)

c_γ0 Thenwpγq 0, apγq 0.

(2)

c_γ 1

m_je_ξP_I Thenwpγq m_j, apγq 1.

(3)

c_γ e_η 1

m_je_ξP_I Thenwpγq m_j, apγq apηq 1¤n_j.

There are three types of functionals.

LetγP∆_{q 1}. (1)

c_γ0 Thenwpγq 0, apγq 0.

(2)

c_γ 1

m_je_ξP_I Thenwpγq m_j, apγq 1.

(3)

c_γ e_η 1

m_je_ξP_I Thenwpγq m_j, apγq apηq 1¤n_j.

There are three types of functionals.

LetγP∆_{q 1}. (1)

c_γ0 Thenwpγq 0, apγq 0.

(2)

c_γ 1

m_je_ξP_I Thenwpγq m_j, apγq 1.

(3)

c_γ e_η 1

m_je_ξP_I Thenwpγq m_j, apγq apηq 1¤n_j.

There are three types of functionals.

LetγP∆_{q 1}. (1)

c_γ0 Thenwpγq 0, apγq 0.

(2)

c_γ 1

m_je_ξP_I Thenwpγq m_j, apγq 1.

(3)

c_γ e_η 1

m_je_ξP_I Thenwpγq m_j, apγq apηq 1¤n_j.

There are three types of functionals.

LetγP∆_{q 1}. (1)

c_γ0 Thenwpγq 0, apγq 0.

(2)

c_γ 1

m_je_ξP_I Thenwpγq m_j, apγq 1.

(3)

c_γ e_η 1

m_je_ξP_I Thenwpγq m_j, apγq apηq 1¤n_j.

∪<sup>`</sup>∈I<sub>1</sub>∆` ∪`∈I2∆` ∪<sup>`</sup>∈I<sub>k</sub>∆`

ξ1 η1 ξ₂ η₂ ξ_k ηk=γ

Forγ P∆_{q 1}withwpγq _m¹_j andαpγq k ¤n_j, there exist ξ1, ξ2, . . . , ξk,η1, η2, . . . , ηk andI₁,I₂, . . . ,I_k such that

e_γ 1 m_j

¸k i1

e_ξ

i P_Ii

¸k i1

d_η

i.

∪<sup>`</sup>∈I<sub>1</sub>∆` ∪`∈I2∆` ∪<sup>`</sup>∈I<sub>k</sub>∆`

ξ1 η1 ξ₂ η₂ ξ_k ηk=γ

Forγ P∆_{q 1}withwpγq _m¹_j andαpγq k ¤n_j, there exist ξ1, ξ2, . . . , ξk,η1, η2, . . . , ηk andI₁,I₂, . . . ,I_k such that

e_γ 1 m_j

¸k i1

e_ξ

i P_Ii

¸k i1

d_η

i.

The mixed Tsirelson BD-L8 space is defined by induction by setting∆1Γ1 tγ1uwithwpγ1q.

Assuming that∆₁, . . . ,∆qhave been defined,∆_{q 1}is defined by taking all possibleφP`₁pΓqqof the previous form withwpφq m_j,j ¤q 1.

It is shown that the extensions operators iq:`<sub>8</sub>pΓqq Ñ`₈pΓqsatisfy}iq} ¤1 8{m₁.

The mixed Tsirelson BD-L8 space is defined by induction by setting∆1Γ1 tγ1uwithwpγ1q.

Assuming that∆₁, . . . ,∆qhave been defined,∆_{q 1}is defined by taking all possibleφP`₁pΓqqof the previous form withwpφq m_j,j ¤q 1.

It is shown that the extensions operators iq:`<sub>8</sub>pΓqq Ñ`₈pΓqsatisfy}iq} ¤1 8{m₁.

The mixed Tsirelson BD-L8 space is defined by induction by setting∆1Γ1 tγ1uwithwpγ1q.

Assuming that∆₁, . . . ,∆qhave been defined,∆_{q 1}is defined by taking all possibleφP`₁pΓqqof the previous form withwpφq m_j,j ¤q 1.

It is shown that the extensions operators iq:`<sub>8</sub>pΓqq Ñ`₈pΓqsatisfy}iq} ¤1 8{m₁.

The mixed Tsirelson BD-L8 space is defined by induction by setting∆1Γ1 tγ1uwithwpγ1q.

Assuming that∆₁, . . . ,∆qhave been defined,∆_{q 1}is defined by taking all possibleφP`₁pΓqqof the previous form withwpφq m_j,j ¤q 1.

It is shown that the extensions operators iq:`<sub>8</sub>pΓqq Ñ`₈pΓqsatisfy}iq} ¤1 8{m₁.

Theorem

The mixed Tsirelson BD-L8 spaceXhas separable dual and does not containc₀or`pfor 1 p  8. Every skipped block sequence with respect to the FDDti<sub>q</sub>p`₈p∆_qqqu⁸_n0generates a reflexive subspace. Hence, every skipped block sequence does not generate aL8space.

Theorem

The mixed Tsirelson BD-L8 spaceXhas separable dual and does not containc₀or`pfor 1 p  8. Every skipped block sequence with respect to the FDDti<sub>q</sub>p`₈p∆_qqqu⁸_n0generates a reflexive subspace. Hence, every skipped block sequence does not generate aL8space.

Theorem

There exists aL8spaceXwith a basispd_nqnsuch that for every infinite subsetLofNthe subspaceX_L  d_n:nPL¡ contains aL8space.

Theorem (A first Step)

There exists a Mixed Tsirelson BDL8space not containingc₀ or`p for 1¤p  8and a skipped block basis that that

generates aL8space.

The definition of this space requires a modification of the previous techniques.

Theorem

There exists aL8spaceXwith a basispd_nqnsuch that for every infinite subsetLofNthe subspaceX_L  d_n:nPL¡ contains aL8space.

Theorem (A first Step)

There exists a Mixed Tsirelson BDL8space not containingc₀ or`p for 1¤p  8and a skipped block basis that that

generates aL8space.

The definition of this space requires a modification of the previous techniques.

Theorem

There exists aL8spaceXwith a basispd_nqnsuch that for every infinite subsetLofNthe subspaceX_L  d_n:nPL¡ contains aL8space.

Theorem (A first Step)

There exists a Mixed Tsirelson BDL8space not containingc₀ or`p for 1¤p  8and a skipped block basis that that

generates aL8space.

The definition of this space requires a modification of the previous techniques.

Theorem

There exists aL8spaceXwith a basispd_nqnsuch that for every infinite subsetLofNthe subspaceX_L  d_n:nPL¡ contains aL8space.

Theorem (A first Step)

There exists a Mixed Tsirelson BDL8space not containingc₀ or`p for 1¤p  8and a skipped block basis that that

generates aL8space.

The definition of this space requires a modification of the previous techniques.

How to create a BDL8space with a Schauder basispdnqn

and a block sequencepx_kqk such that x_k :k PN¡is also aL8space.

We start with a sequence of parameterspm_j,n_jqj as before.

For define a setΓ Y⁸_n1∆q such that each∆q tγuand ifγ P∆_2q1then

wpγq 0ôc_γ 0.

How to create a BDL8space with a Schauder basispdnqn

and a block sequencepx_kqk such that x_k :k PN¡is also aL8space.

We start with a sequence of parameterspm_j,n_jqj as before.

For define a setΓ Y⁸_n1∆q such that each∆q tγuand ifγ P∆_2q1then

wpγq 0ôc_γ 0.

How to create a BDL8space with a Schauder basispdnqn

and a block sequencepx_kqk such that x_k :k PN¡is also aL8space.

We start with a sequence of parameterspm_j,n_jqj as before.

For define a setΓ Y⁸_n1∆q such that each∆q tγuand ifγ P∆_2q1then

wpγq 0ôc_γ 0.

How to create a BDL8space with a Schauder basispdnqn

and a block sequencepx_kqk such that x_k :k PN¡is also aL8space.

We start with a sequence of parameterspm_j,n_jqj as before.

For define a setΓ Y⁸_n1∆q such that each∆q tγuand ifγ P∆_2q1then

wpγq 0ôc_γ 0.

Ifγ P∆_2q thenwpγq pm_2i,m_2j11, . . . ,m_2jk1q, where 2i ¤2q,

1¤k ¤n_2i,m_2jk1is determined bypm_2i1, . . . ,m_2jk11q. In this case we say thatγ hasmultiple weightandm_2i is called itsprimitive weight.

Ifγ P∆_2q thenwpγq pm_2i,m_2j11, . . . ,m_2jk1q, where 2i ¤2q,

1¤k ¤n_2i,m_2jk1is determined bypm_2i1, . . . ,m_2jk11q. In this case we say thatγ hasmultiple weightandm_2i is called itsprimitive weight.

Ifγ P∆_2q thenwpγq pm_2i,m_2j11, . . . ,m_2jk1q, where 2i ¤2q,

1¤k ¤n_2i,m_2jk1is determined bypm_2i1, . . . ,m_2jk11q. In this case we say thatγ hasmultiple weightandm_2i is called itsprimitive weight.

Ifγ P∆_2q thenwpγq pm_2i,m_2j11, . . . ,m_2jk1q, where 2i ¤2q,

1¤k ¤n_2i,m_2jk1is determined bypm_2i1, . . . ,m_2jk11q. In this case we say thatγ hasmultiple weightandm_2i is called itsprimitive weight.

Ifγ P∆_2q thenwpγq pm_2i,m_2j11, . . . ,m_2jk1q, where 2i ¤2q,

1¤k ¤n_2i,m_2jk1is determined bypm_2i1, . . . ,m_2jk11q. In this case we say thatγ hasmultiple weightandm_2i is called itsprimitive weight.

The idea in the use of the multiple weights is to succeed that theγ which codesc_γ, at the same time norms a block vector.

How to define aη with multiple weightswpηq, starting with γ withwpγq pm_2i,m_2j11, . . . ,m_2j`1q:

Ifγ PΓ_k, chooseξ PΓnzΓ_k and setI pk,ns.

The idea in the use of the multiple weights is to succeed that theγ which codesc_γ, at the same time norms a block vector.

How to define aη with multiple weightswpηq, starting with γ withwpγq pm_2i,m_2j11, . . . ,m_2j`1q:

Ifγ PΓ_k, chooseξ PΓnzΓ_k and setI pk,ns.