Order preserving and order reversing operators on the class of

Order preserving and order reversing operators onthe class of convex functions in Banach spaces

Daniel Reem

Joint work with Alfredo N. Iusem and Benar F. Svaiter

IMPA (Rio, Brazil)

http://w3.impa.br/~dream

May 29, 2013

Banach Spaces: Geometry and Analysis,A conference in memory of Joram Lindenstrauss,

Institute for Advanced Studies,Hebrew University, Jerusalem, Israel

Iusem, Reem, Svaiter (IMPA) Order preserving and reversing operators May 29, 2013 1 / 23

Historical background: the birth of this work

Feb 2009, The Technion, room 232: a conference in honor ofYehoram Gordon: Probability and Geometry of Convex Sets.

I was a Ph.D student there. I worked in different domains but walkedto some talks.

One of the talks: of Vitali Milman. Spoke on his results with ShiriArtstein-Avidan.

Attracted me: perhaps because I did something related to orderreversing operators in a different context (computer science and fixedpoint theory).

Interesting talk. At the end I asked: is anything known in the infinitedimensional case?

Milman said:§ interesting question which has been asked several times in the past;

§ the answer is: nothing is known.

Feb 2009, The Technion,

room 232: a conference in honor ofYehoram Gordon: Probability and Geometry of Convex Sets.

Feb 2009, The Technion, room 232:

a conference in honor ofYehoram Gordon: Probability and Geometry of Convex Sets.

Milman said:

§ interesting question which has been asked several times in the past;

Historical background (Cont.)

End of August 2011: I arrived at IMPA (Brazil) for a postdoc.

Slowly discovered that there are some very strong mathematiciansthere.

June 2012: took a careful look at the paper of Artstein-Avidan andMilman.

July 2012: suggested to Alfredo N. Iusem and Benar F. Svaiter towork on a joint project.

December 2012: a related preprint was uploaded onto the arXiv.

Mathematical background: convex analysis (Rn)

Notation: C pRnq is the set of lower semicontinuous proper convexfunctions f : Rn Ñ RY t`8u.

Reminder: A function f : Rn Ñ RY t`8u is called proper wheneverit is not identically 8;

Fenchel conjugation (convex conjugate, Legendretransform)

Definition

Given f P C pRnq, its conjugation is the function f ˚ : Rn Ñ p´8,8sdefined by

f ˚px˚q “ suptxx˚, xy ´ f pxq : x P Rnu, @x˚ P Rn

where xx˚, xy is the inner product of x˚ and x.

Fenchel conjugation (convex conjugate, Legendretransform)

Definition

Given f P C pRnq, its conjugation is the function f ˚ : Rn Ñ p´8,8sdefined by

f ˚px˚q “ suptxx˚, xy ´ f pxq : x P Rnu, @x˚ P Rn

where xx˚, xy is the inner product of x˚ and x.

Convex analysis on Rn: some facts

The Legendre transform appears in many scientific domains

the operator F : C pRnq Ñ C pRnq, F pf q “ f ˚, is fully order reversing:

§ It is invertible,

§ it reverses the (pointwise) order: f ď g ùñ F pf q ě F pgq,

§ its inverse also reverses the order: h ď k ùñ F ´1phq ě F ´1pkq

Actually, F ´1 “ F (since f “ f ˚˚)

The results of Artstein-Avidan & Milman (2007, 2009)

Theorem

An operator T : C pRnq Ñ C pRnq is fully order preserving if and only ifthere exist c P Rn, w P Rn, β P R, τ ą 0 and an invertible linear operatorE : Rn Ñ Rn such that

T pf qpxq “ τ f pEx ` cq ` xw , xy ` β,

for all f P C pRnq and all x P Rn.

Loosely speaking: up to linear terms T is the identity operator.

Theorem

Loosely speaking:

up to linear terms T is the identity operator.

Theorem

The results of AA-M (Cont.)

Theorem

An operator T : C pRnq Ñ C pRnq is fully order reversing if and only ifthere exist v0 P Rn, v1 P Rn, ρ P R, τ ą 0 and an invertible linear operatorH : Rn Ñ Rn such that

T pf qpxq “ τ f ˚pHx ` v0q ` xv1, xy ` ρ,

Loosely speaking: up to linear terms T is the Legendre transform.

Theorem

Loosely speaking:

up to linear terms T is the Legendre transform.

Theorem

An operator T : C pRnq Ñ C pRnq is order reversing and satisfiesT ˝ T “ id if and only if there exist C0 P R, v0 P Rn, and an invertiblesymmetric linear operator B : Rn Ñ Rn such that

pTf qpxq “ f ˚pBx ` v0q ` xv0, xy ` C0,

Theorem

Loosely speaking:

up to linear terms T is the Legendre transform.

Theorem

The results of AA-M: a discussion

Unexpected:§ only very few assumptions are imposed on the operator;

§ it acts on a rich class of functions;

§ but despite this, its form is fully characterized.

overlooked fact: a quite fundamental fact that has not beenobserved before.

An abstract form of duality: Let Λ be a class of functions definedon Rn. An operator T : Λ Ñ Λ is called a duality transform if

§ T is an involution: T pT pf qq “ f for all f P Λ.

§ T is order reversing.

Their motivation was to show that:§ such a notion appears in central examples in analysis and geometry

§ the two simple assumptions suffice to characterize the form of T .

Unexpected:

§ only very few assumptions are imposed on the operator;

Their motivation was to show that:

§ such a notion appears in central examples in analysis and geometry

Later developments

Several people, mainly from Tel-Aviv University, have shown concreteform for other operators acting on other classes of objects

A few involved names: Alesker, Artstein-Avidan, Boroczky,Faifman, Florentin, Milman, Segal, Slomka, Schneider

A Rich theory. However, it is inherently finite dimensional

Later developments

A Rich theory.

However, it is inherently finite dimensional

Later developments

Previous developments?

1 Somewhat related results regarding characterization of fully orderpreserving operators (possibly having additional features) acting oncertain (rigid) structures (e.g., ordered cones)

2 Related to physics: special relativity, quantum mechanics, etc.

3 Some names: Alexandrov, Kadison, Molnar, Noll, Ovcinnikova,Rothaus, Schaffer, Zeeman

4 Nothing about the class of convex functions

back to convex functions: going beyond finitedimension

C pX q is naturally defined when X is an infinite dimensional space (we justreplace Rn by X ) but some difficulties occur, including:

(1) Issues related to the conjugation f ÞÑ f ˚ (not necessarily invertible)

(2) Issues related to the canonical embedding of X in the bidual X ˚˚

(3) Issues related to the biconjugate f ˚˚

(4) Issues related to linear operators (not automatically continuous)

(5) Issues related to topologies (weak, weak star, and strong)

dimension infinity: any known related result?

To the best of our knowledge, only one, by Stephen Wright (2010):

Proved an interesting result concerning locally convex Hausdorfftopological vector spaces X (with a compatible topology on X ˚).

But: No characterization of the kind established by AA-M was given.

8-dimensional convex analysis: convex conjugate

Let X be a real Banach space,

Let X ˚ be its dual.

Definition

Given f P C pX q, its Fenchel conjugation is the functionf ˚ : X ˚ Ñ p´8,8s defined by

f ˚px˚q “ suptxx˚, xy ´ f pxq : x P X u,

where xx˚, xy “ x˚pxq is the operation of x˚ P X ˚ on x P X .

Definition

8-dimensional convex analysis (Cont.)

The operator F : C pX q Ñ C pX ˚q, F pf q “ f ˚, is order reversing

The operator F is invertible if and only if X is reflexive.

However, if Cw˚pX ˚q denotes the set of weak˚ lower semicontinuousproper convex functions from X ˚ to RY t`8u, then now the Fenchelconjugation F : C pX q Ñ Cw˚pX ˚q, F pf q “ f ˚ is invertible and fullyorder reversing. Its inverse is

F ´1pgqpxq “ suptxx , x˚y ´ gpx˚q : x˚ P X ˚u

for all g P Cw˚pX ˚q and all x P X (where xx , x˚y “ x˚pxq).

Our results (2014?)

In all of our results X is a real Banach space of dimension at least 2.

Our results (2014?)

In all of our results X is a real Banach space of dimension at least 2.

Our results: order preserving

Theorem

An operator T : C pX q Ñ C pX q is fully order preserving if and only if thereexist c P X , w P X ˚, β P R, τ ą 0 and a continuous invertible linearoperator E : X Ñ X such that

for all f P C pX q and all x P X .

Theorem

Loosely speaking:

Theorem

Our results: order reversing

Theorem

An operator S : C pX q Ñ Cw˚pX ˚q is fully order reversing if and only ifthere exist v P X ˚, y P X , ρ P R, τ ą 0 and a (norm) continuousinvertible linear operator H : X ˚ Ñ X ˚ such that

Spf qpx˚q “ τ f ˚pHx˚ ` vq ` xx˚, yy ` ρ,

for all f P C pX q and all x˚ P X ˚.

Loosely speaking: up to linear terms S is the Fenchel conjugation.

Theorem

Loosely speaking:

up to linear terms S is the Fenchel conjugation.

Theorem

Our results: involutions

Theorem

An operator T : C pX q Ñ C pX q is an order preserving involution if andonly if there exist c P X , w P X ˚ and a continuous invertible linearoperator E : X Ñ X satisfying E 2 “ IX , c P KerpE ` IX q,w P KerpE ˚ ` IX˚q, such that

T pf qpxq “ f pEx ` cq ` xw , xy ´1

2xc ,wy

for all f P C pX q and all x P X , where IX is the identity operator in X .

Theorem

2xc ,wy

Theorem

2xc ,wy

Loosely speaking:

Theorem

2xc ,wy

Comparison between our approach to AA-M

Similarities:

Formulation of basic properties based on order considerations

Use of tools from convex analysis

Working with specific subclasses first

Use of a lemma from affine geometry

Similarities:

Comparison between our approach to AA-M (Cont.)

Differences:

A dual approach: we establish the order preserving result first andthen conclude the order reversing one; they do the opposite.

Infinite dimensional considerations are taken into account; forinstance, to show that the linear operator E in the expression of Tf

T pf qpxq “ τ f pEx ` cq ` xw , xy ` β.

is continuous (otherwise Tf may not be l.s.c)

Properness is used explicitly and implicitly in several key places (theyuse it implicitly)

We obtain some results, e.g., about characterizations of affinefunctions, which are possibly interesting on their own right.

Differences:

Infinite dimensional considerations are taken into account;

forinstance, to show that the linear operator E in the expression of Tf

Differences:

The End

arXiv preprint: http://arxiv.org/abs/1212.1120

related slideshow: http://w3.impa.br/~dream

The End

Order preserving and order reversing operators on the class of

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