luiza a. moraes · luiza a. moraes instituto de matemática universidade federal do rio de janeiro...

Tauberian Polynomials

Luiza A. Moraes

Instituto de MatemáticaUniversidade Federal do Rio de Janeiro

Workshop on Functional Analysis Valencia 2013on the occasion of the 60th birthday of Andreas Defant

L. Moraes (UFRJ) Tauberian Polynomials Valencia, 06/06/2013 1 / 19

All the results presented in this talk were obtained in collaboration with MariaD. Acosta (Universidad de Granada) and Pablo Galindo (Universidad deValência).

Tauberian Operators

X , Y = real or complex Banach spaces.

• Known: a continuous linear operator T : X → Y is weakly compact ifT ∗∗(X ) ⊂ Y (where T ∗∗ denotes the second adjoint of T ).

DefinitionA continuous linear operator T : X → Y is Tauberian if T ∗∗−1(Y ) ⊂ X (i.e.,T ∗∗(X ′′ \ X ) ⊂ Y ′′ \ Y ).

Examples• If Z is a closed subspace of X :(1) the canonical immersion J : Z → X is a Tauberian operator;(2) the quotient operator QZ : X → X/Z is Tauberian if and only if Z is

reflexive.

• The canonical immersion ι of the James space J in c0 is a Tauberianoperator.

Tauberian OperatorsX , Y = real or complex Banach spaces.

reflexive.

Examples• If Z is a closed subspace of X :(1) the canonical immersion J : Z → X is a Tauberian operator;

(2) the quotient operator QZ : X → X/Z is Tauberian if and only if Z isreflexive.

reflexive.

Tauberian Operators

N. Kalton and A. Wilansky studied these operators in Tauberian operatorson Banach spaces, Proc. Amer. Math. Soc. 57 (1976), and refered to themas Tauberian operators.

We also refer to the recent monograph:M. González and A. Martínez-Abejó, Tauberian Operators, OperatorTheory: Advances and Applications 194 Birkhäuser Verlag, Basel (2010).

Some applications of Tauberian operators in the theory of Banach spaces:

• in the proof of the Davies, Figiel, Johnson and Pelczynski FactorizationTheorem [cf. Factoring weakly compact operators, J. Funct. Anal. 17(1974)].

• in the proof of the equivalence between the Radon-Nikodym property andthe Krein-Milman property given by Schachermayer [cf. For a Banach spaceisomorphic to its square the Radon-Nikodým property and theKrein-Milman property are equivalent, Studia Math. 81 (1985)].

Tauberian Operators

Homogeneous Polynomials

Notation:

P(NX , Y ) = space of the continuous N-homogeneous polynomials from Xinto Y .

P(NX ) = P(NX , K), where K = R or C.

P(X , Y ) = space of all continuous polynomials from X into Y .

‖P‖ = sup{‖P(x)‖ : ‖x‖ ≤ 1}.

Pwu(NX , Y ) = space of all P ∈ P(NX , Y ) whose restrictions to the boundedsubsets of X are weakly continuous.

Aron and Berner showed that to each P ∈ P(NX , Y ) corresponds aP ∈ P(NX ′′, Y ′′) such that P|X = P.

Davie and Gamelin showed (10 years latter) that ‖P‖ = ‖P‖.

Notation:

‖P‖ = sup{‖P(x)‖ : ‖x‖ ≤ 1}.

Notation:

‖P‖ = sup{‖P(x)‖ : ‖x‖ ≤ 1}.

Notation:

‖P‖ = sup{‖P(x)‖ : ‖x‖ ≤ 1}.

Notation:

‖P‖ = sup{‖P(x)‖ : ‖x‖ ≤ 1}.

Notation:

‖P‖ = sup{‖P(x)‖ : ‖x‖ ≤ 1}.

Notation:

‖P‖ = sup{‖P(x)‖ : ‖x‖ ≤ 1}.

DefinitionWe say that P ∈ P(NX , Y ) is Tauberian if P satisfies P(X ′′ \ X ) ⊂ Y ′′ \ Y or,equivalently, P−1(Y ) ⊂ X .

Examples• X reflexive ⇒ P ∈ P(NX , Y ) is Tauberian for all Y and N ∈ N.

• Y reflexive and X non reflexive ⇒ every P ∈ P(NX , Y ) is not a Tauberianpolynomial.

• ι2 : J → c0 given by ι2(x) := (ι(x))2 is a Tauberian polynomial (where ι isthe canonical embedding of the James space J into c0).

• X non reflexive and P : X → Y weakly compact N-homogeneouspolynomial ⇒ P is not Tauberian. In particular, every element of Pwu(NX , Y )is not a Tauberian polynomial.

Polynomials versus the symmetric tensor product

⊗N,s,πX = the completion of the N-fold symmetric tensor product of X ,endowed with the projective norm.

Well known:• ⊗N,s,πX is a Banach space whose topological dual is linearly isomorphic tothe space P(NX ).

• L(⊗N,s,πX , Y ) is linearly isomorphic to the space P(NX , Y ).

More explicitly, each P ∈ P(NX , Y ) can be identified with a linear operatorLP ∈ L(⊗N,s,πX , Y ) such that P(x) = LP(x ⊗ · · · ⊗ x) ∀x ∈ X .

We say that LP is the linearization of P.

If δ : X → ⊗N,s,πX is the N-homogeneous polynomial given byδ(x) = x ⊗ · · · ⊗ x , we have P = LP ◦ δ for every P ∈ P(NX , Y ).

Well known:

• ⊗N,s,πX is a Banach space whose topological dual is linearly isomorphic tothe space P(NX ).

ExampleThe mapping δ : x ∈ X → x ⊗ · · · ⊗ x ∈ ⊗N,s,πX is a Tauberian polynomial

Indeed, let

δ : X ′′ → (⊗N,s,πX )′′ be the Aron-Berner extension of δ.

LP ◦ δ = P ⇒ δ(x ′′)(LP) = LP ◦ δ(x ′′) = P(x ′′) ∀x ′′ ∈ X ′′ and ∀P ∈ P(NX )

Claim: x ′′ ∈ X ′′\{0} : δ(x ′′) ∈ ⊗N,s,πX ⇒ x ′′ ∈ X

It is enough to show that x ′′ is w*-continuous on BX ′

Take a net (x ′α) ⊂ BX ′ that converges to x ′ ∈ BX ′ in the w*-topology of X ′ andy ′ ∈ X ′ such that x ′′(y ′) 6= 0.

Let P = (y ′)N−1x ′ ∈ P(NX ) and, for each α, Pα = (y ′)N−1x ′α ∈ P(NX ).

Indeed, let

Tauberian PolynomialsThe bounded net (Pα) converges pointwise to P on X .

i.e., LPα(x ⊗ · · · ⊗ x) → LP(x ⊗ · · · ⊗ x) ∀x ∈ X .

Now:(LPα

)bounded in

(⊗N,s,πX

)′ and span{x ⊗ · · · ⊗ x : x ∈ X} norm-dense in⊗N,s,πX ⇒ LPα

→ LP in the w ∗ −topology of(⊗N,s,πX

)′.δ(x ′′) ∈ ⊗N,s,πX (by hypothesis) ⇒ δ(x ′′)(LPα

) → δ(x ′′)(LP).

δ(x ′′)(LPα) = Pα(x ′′) and δ(x ′′)(LP) = P(x ′′) ⇒ Pα(x ′′) → P(x ′′).

Pα and P are finite products of functionals ⇒ Pα(x ′′) = x ′′(y ′)N−1x ′′(x ′α)and P(x ′′) = x ′′(y ′)N−1x ′′(x ′).Hence,

x ′′(y ′)N−1x ′′(x ′α) → x ′′(y ′)N−1x ′′(x ′).From this and from x ′′(y ′) 6= 0 we obtain

x ′′(x ′α) → x ′′(x ′) .

We have shown that x ′′ is w∗-continuous on BX ′ . Therefore, x ′′ ∈ X .

i.e., LPα(x ⊗ · · · ⊗ x) → LP(x ⊗ · · · ⊗ x) ∀x ∈ X .

Now:(LPα

)bounded in

(⊗N,s,πX

) → δ(x ′′)(LP).

x ′′(x ′α) → x ′′(x ′) .

i.e., LPα(x ⊗ · · · ⊗ x) → LP(x ⊗ · · · ⊗ x) ∀x ∈ X .

Now:(LPα

)bounded in

(⊗N,s,πX

δ(x ′′) ∈ ⊗N,s,πX (by hypothesis) ⇒ δ(x ′′)(LPα) → δ(x ′′)(LP).

x ′′(x ′α) → x ′′(x ′) .

i.e., LPα(x ⊗ · · · ⊗ x) → LP(x ⊗ · · · ⊗ x) ∀x ∈ X .

Now:(LPα

)bounded in

(⊗N,s,πX

) → δ(x ′′)(LP).

x ′′(x ′α) → x ′′(x ′) .

i.e., LPα(x ⊗ · · · ⊗ x) → LP(x ⊗ · · · ⊗ x) ∀x ∈ X .

Now:(LPα

)bounded in

(⊗N,s,πX

) → δ(x ′′)(LP).

x ′′(x ′α) → x ′′(x ′) .

i.e., LPα(x ⊗ · · · ⊗ x) → LP(x ⊗ · · · ⊗ x) ∀x ∈ X .

Now:(LPα

)bounded in

(⊗N,s,πX

) → δ(x ′′)(LP).

Pα and P are finite products of functionals ⇒ Pα(x ′′) = x ′′(y ′)N−1x ′′(x ′α)and P(x ′′) = x ′′(y ′)N−1x ′′(x ′).

Hence,

x ′′(x ′α) → x ′′(x ′) .

i.e., LPα(x ⊗ · · · ⊗ x) → LP(x ⊗ · · · ⊗ x) ∀x ∈ X .

Now:(LPα

)bounded in

(⊗N,s,πX

) → δ(x ′′)(LP).

x ′′(x ′α) → x ′′(x ′) .

i.e., LPα(x ⊗ · · · ⊗ x) → LP(x ⊗ · · · ⊗ x) ∀x ∈ X .

Now:(LPα

)bounded in

(⊗N,s,πX

) → δ(x ′′)(LP).

x ′′(x ′α) → x ′′(x ′) .

i.e., LPα(x ⊗ · · · ⊗ x) → LP(x ⊗ · · · ⊗ x) ∀x ∈ X .

Now:(LPα

)bounded in

(⊗N,s,πX

) → δ(x ′′)(LP).

x ′′(x ′α) → x ′′(x ′) .

i.e., LPα(x ⊗ · · · ⊗ x) → LP(x ⊗ · · · ⊗ x) ∀x ∈ X .

Now:(LPα

)bounded in

(⊗N,s,πX

) → δ(x ′′)(LP).

x ′′(x ′α) → x ′′(x ′) .

We have shown that x ′′ is w∗-continuous on BX ′ . Therefore, x ′′ ∈ X .L. Moraes (UFRJ) Tauberian Polynomials Valencia, 06/06/2013 9 / 19

PropositionAssume that P ∈ P(NX , Y ) is a Tauberian polynomial and T ∈ L(Y , Z ) is aTauberian operator, then T ◦ P is a Tauberian polynomial. A partial converseholds: if T ◦ P is a Tauberian polynomial, then P itself is Tauberian.

CorollaryIf the linearization of P ∈ P(NX , Y ) is Tauberian, then P itself is Tauberian.

However:

P = Tauberian polynomial ; its linearization TP is Tauberian.

Example:Recall that the quotient operator QZ : X → X/Z is Tauberian ⇔ Z isreflexive.

However:

Example:

Recall that the quotient operator QZ : X → X/Z is Tauberian ⇔ Z isreflexive.

However:

Hence:

`2⊗s,π`2 contains a copy of `1 ⇒ the quotient mapping

q : `2⊗s,π`2 →`2⊗s,π`2

is not a Tauberian operator.

However, the polynomial

x ∈ `2 7→ P(x) := x ⊗ x + `1 ∈`2⊗s,π`2

is Tauberian (since `2 is reflexive) and

P(x) = q(x ⊗ x) ⇒ q is the linearization of P.

Hence:

q : `2⊗s,π`2 →`2⊗s,π`2

x ∈ `2 7→ P(x) := x ⊗ x + `1 ∈`2⊗s,π`2

Hence:

q : `2⊗s,π`2 →`2⊗s,π`2

x ∈ `2 7→ P(x) := x ⊗ x + `1 ∈`2⊗s,π`2

Hence:

q : `2⊗s,π`2 →`2⊗s,π`2

x ∈ `2 7→ P(x) := x ⊗ x + `1 ∈`2⊗s,π`2

Tauberian Operators ( Known characterizations)

PropositionIf T : X → Y is linear and continuous, the following are equivalent:(1) T ∗∗(X ′′ \ X ) ⊂ Y ′′ \ Y .

(2) If B ⊂ X is a bounded set such that T (B) is weakly relatively compact,then B is weakly relatively compact.

(3) If B ⊂ X is a bounded set such that T (B) is relatively compact, then B isweakly relatively compact.

PropositionIf X is a weakly sequentially complete Banach space, Y is an arbitrary Banachspace and T : X → Y is linear and continuous, the following are equivalents:(1) T ∗∗(X ′′ \ X ) ⊂ Y ′′ \ Y .

(2) (T ∗∗)−1(0) = T−1(0).

Tauberian Operators ( Known characterizations)

PropositionIf T : X → Y is linear and continuous, the following are equivalent:(1) T ∗∗(X ′′ \ X ) ⊂ Y ′′ \ Y .

(2) If B ⊂ X is a bounded set such that T (B) is weakly relatively compact,then B is weakly relatively compact.

(3) If B ⊂ X is a bounded set such that T (B) is relatively compact, then B isweakly relatively compact.

PropositionIf X is a weakly sequentially complete Banach space, Y is an arbitrary Banachspace and T : X → Y is linear and continuous, the following are equivalents:(1) T ∗∗(X ′′ \ X ) ⊂ Y ′′ \ Y .

(2) (T ∗∗)−1(0) = T−1(0).

Because of the lack of linearity, some difficulties arise:

• Given P ∈ P(NX , Y ), we can’t say in general that P(X ) is a linear subspaceof Y .

• In general it is not true that P(A) is a convex subset of Y whenever A is aconvex subset of X .

• The equality T ∗∗(Aw∗

) = T (A)w∗

for all bounded and convex subset A of Ecannot be extended to the case of polynomials (since in general P(A) is notconvex).

Consequently, the weak topology can not play the same role as in the linearsetting.

) = T (A)w∗

The polynomial topologies τp and τp∗

The polynomial topology τp (resp. τpN ) on X is the smallest topology forwhich a net (xα) converges to x if and only if P(xα) → P(x) ∀P ∈ P(X )(resp. ∀P ∈ P(mX ) para todo m ≤ N ∈ N).

The polynomial-star topology τp∗ (respectively, τp∗N ) on X ′′ is the smallesttopology for which a net (zα) converges to z if and only ifP(zα) → P(z) ∀P ∈ P(X ) (resp. ∀P ∈ P(mX ) para todo m ≤ N ∈ N).

Theorem (Davie and Gamelin)Let S be a bounded convex subset of X . Then the weak*-closure of S in X ′′

coincides with the polynomial-star closure of S in X ′′.

There is not a general Banach Alaoglu Polynomial Theorem, but we showedthe following:

Banach Alaoglu Polynomial TheoremIf X is a Banach space, then Pwu(NX ) = P(NX ) if and only if every boundedand τp∗N -closed subset A of X ′′ is compact in the τp∗N -topology. A similarstatement holds for the equality P(X ) = Pwu(X ) and the τp∗ -topology.

TeoremaLet y ∈ Y , and P ∈ P(NX , Y ). The following statements are equivalent:(a) P−1(y) ⊂ X .

(b) If (xα)α∈Λ is a net in X such that P(xα)w→ y , then every τp∗N -cluster point

of(xα

)α∈Λ

belongs to X .

CorollaryLet P ∈ P(k X , Y ). The following statements are equivalent:(a) N(P) = N(P).

(b) If (xα) is a net in X such that P(xα)w→ 0, then every τp∗k -cluster point of(

)belongs to X .

TeoremaLet y ∈ Y , and P ∈ P(NX , Y ). The following statements are equivalent:(a) P−1(y) ⊂ X .

(b) If (xα)α∈Λ is a net in X such that P(xα)w→ y , then every τp∗N -cluster point

of(xα

)α∈Λ

belongs to X .

CorollaryLet P ∈ P(k X , Y ). The following statements are equivalent:(a) N(P) = N(P).

(b) If (xα) is a net in X such that P(xα)w→ 0, then every τp∗k -cluster point of(

)belongs to X .

TheoremLet P ∈ P(NX , Y ). Consider the following statements:(a) P is Tauberian.(b) P(BX ) is weakly closed and x ′′ ∈ X whenever x ∈ X , x ′′ ∈ X ′′ and

P(x ′′) = P(x).

(c) P(BX ) is weakly closed and N(P) = N(P).

Then (b) ⇒ (a) and (c), and whenever Pwu(NX ) = P(NX ), (a) ⇒ (b).

In general it may happen that the image of a non τpN -relatively compactsubset under a Tauberian polynomial is weakly relatively compact (we havean example). By assuming extra conditions on the domain space we candeduce compactness of a subset for the τpN -topology.

TheoremLet P ∈ P(NX , Y ). Consider the following statements:(a) P is Tauberian.(b) P(BX ) is weakly closed and x ′′ ∈ X whenever x ∈ X , x ′′ ∈ X ′′ and

P(x ′′) = P(x).

(c) P(BX ) is weakly closed and N(P) = N(P).

Then (b) ⇒ (a) and (c), and whenever Pwu(NX ) = P(NX ), (a) ⇒ (b).

In general it may happen that the image of a non τpN -relatively compactsubset under a Tauberian polynomial is weakly relatively compact (we havean example). By assuming extra conditions on the domain space we candeduce compactness of a subset for the τpN -topology.

TheoremLet P ∈ P(NX , Y ) be a Tauberian polynomial. If C is a bounded subset of Xsuch that P(C) is weakly relatively compact, then the closure of C for theτp∗N -topology lies in X . If moreover, P(NX ) = Pwu(NX ), then C is τpN -relativelycompact.

For separable Banach spaces we have

TheoremLet X be separable and P(NX ) = Pwu(NX ). The polynomial P ∈ P(NX , Y ) isTauberian if and only if every bounded subset C of X such that P(C) is weaklyrelatively compact is weakly relatively compact.

TheoremLet P ∈ P(NX , Y ) be a Tauberian polynomial. If C is a bounded subset of Xsuch that P(C) is weakly relatively compact, then the closure of C for theτp∗N -topology lies in X . If moreover, P(NX ) = Pwu(NX ), then C is τpN -relativelycompact.

For separable Banach spaces we have

TheoremLet X be separable and P(NX ) = Pwu(NX ). The polynomial P ∈ P(NX , Y ) isTauberian if and only if every bounded subset C of X such that P(C) is weaklyrelatively compact is weakly relatively compact.

Examples:

• X = T ∗J (the Tsirelson-James space)

• X = d∗(w , 1) whenever w /∈ `N . (where d∗(w , 1)denotes the canonicalpre-dual of the Lorentz sequence space d(w , 1), where w is a null sequenceof positive real numbers such that w /∈ `1• X = C(K ) where K is a metrizable dispersed compact Hausdorff topologicalspace.

TheoremLet P ∈ P(NX , Y ). Assume that the weak topology on P(BX )

wis metrizable. If

every bounded subset C of X such that P(C) is weakly relatively compact isτpN -relatively compact, then P is Tauberian and P(BX ) is weakly closed.

Examples:• X = T ∗

J (the Tsirelson-James space)

wis metrizable. If

• X = d∗(w , 1) whenever w /∈ `N . (where d∗(w , 1)denotes the canonicalpre-dual of the Lorentz sequence space d(w , 1), where w is a null sequenceof positive real numbers such that w /∈ `1

• X = C(K ) where K is a metrizable dispersed compact Hausdorff topologicalspace.

wis metrizable. If

REFERÊNCIAS

[1] M. D. Acosta, P. Galindo and L. A. Moraes, Tauberian polynomials,submitted.

[2] M. González and A. Martínez-Abejón, Tauberian Operators, OperatorTheory: Advances and Applications, vol. 194, Birkhäuser Verlag, Basel, 2010.

[3] N.J. Kalton and A. Wilansky, Tauberian operators on Banach spaces, Proc.Amer. Math. Soc. 57 (1976), 251–255.

luiza a. moraes · luiza a. moraes instituto de matemática universidade federal do rio de janeiro...

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