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Prof. Marek Wisła

Adama Mickiewicz University in Poznań, Poland

Positivity VII, Zaanen Centennial Conference,

Leiden July 22-26, 2013, The Netherlands

Photo: Sandra Sardjono

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A linear operator 𝑇 from a Banach space 𝑋 to another Banach sapace 𝑌 is called compact if the image under 𝑇 of any bounded subset of 𝑋 is a relatively compact subset of 𝑌.

Assume that 𝐾 is a compact Hausdorff space. To any linear operator 𝑇: 𝑋 → 𝐶(𝐾, 𝑅) we can associate a continuous function 𝑇∗: 𝐾 → 𝑋∗ defined by the formula

𝑇∗ 𝑧 𝑥 = 𝑇 𝑥 𝑧 , 𝑧 ∈ 𝐾, 𝑥 ∈ 𝑋.

Positivity VII, Leiden, July 22-26, 2013, The Netherlands 2

A linear operator 𝑇: 𝑋 → 𝐶(𝐾, 𝑅) is called nice if 𝑇∗ 𝐾 ⊂ 𝑒𝑥𝑡 𝐵 𝑋∗ ,

where 𝑒𝑥𝑡 𝐵 𝑋∗ denotes the set of extreme points of the unit ball of the Banach space 𝑋∗.

Blumenthal, Lindenstrauss, Phelps

A compact linear operator from a Banach space 𝑋 into the space of continuous functions 𝐶(𝐾, 𝑅) is extreme provided it is nice.


Blumenthal, Lindenstrauss, Phelps

If 𝑋 is a finite dimensional normed linear space such that the dim𝑋 ≤ 3 or the unit ball 𝐵(𝑋) is plyhedron then {𝑡 ∈ 𝐾: 𝑇∗ 𝑡 ∈ 𝑒𝑥𝑡 𝐵(𝑋∗)} is a dense subset of 𝐾 for every extreme linear operator 𝑇: 𝑋 → 𝐶(𝐾).

B.L.P. gave an example of a four dimensional Banach space 𝑋 and an extreme linear operator 𝑇: 𝑋 → 𝐶[0,1] such that 𝑇∗ 𝑡 ∉ 𝑒𝑥𝑡 𝐵(𝑋∗) for every t ∈ [0,1]


The nice condition can be weakened as long as the set of extreme points e𝑥𝑡 𝐵(𝑋∗) is closed, namely it suffices to assume than

𝑇∗ 𝐾0 ⊂ 𝑒𝑥𝑡 𝐵(𝑋∗)

for some dense subset 𝐾0 ⊂ 𝐾.

Indeed,

𝑇∗ 𝐾 ⊂ 𝑇∗ 𝐾0 ⊂ 𝑒𝑥𝑡 𝐵(𝑋∗).


Characterize those Banach spaces in which the set of extreme points of the unit ball is closed.

Samples

𝐿1: OK, since 𝑒𝑥𝑡 𝐵 𝐿1 = ∅.

𝐿𝑝, 𝑝 > 1: OK.


A function Φ: 𝑅 → 0,∞ is called an Orlicz function, if Φ 0 = 0, Φ is not identically equal to 0, it is even, continuous and convex on the interval (−𝑏Φ, 𝑏Φ) and left-continuous at 𝑏Φ, where 𝑏Φ = sup{𝑢 ≥ 0: Φ 𝑢 < ∞}.

We shall denote 𝑎Φ = sup{𝑢 ≥ 0:Φ 𝑢 = 0}.


By the Orlicz space we mean the space of all Φ –

integrable functions with a constant 𝑘 > 0, i.e.,

𝐼Φ 𝑘𝑥 = Φ 𝑘𝑥(𝑡) 𝑑𝜇𝑇

< ∞ for some 𝑘 > 0.

By p-Amemiya norm we mean the functional

defined by

𝑥 Φ,𝑝 = inf𝑘>0

1

𝑘1 + 𝐼Φ

p𝑘𝑥

1

𝑝, if 1 ≤ 𝑝 < ∞,

𝑥 Φ,∞ = inf𝑘>0

1

𝑘max{1, 𝐼Φ 𝑘𝑥 }, if 𝑝 = ∞.

Positivity VII, Leiden, July 22-26, 2013, The

Netherlands 9

If Φ is an Orlicz function, then the complementary function Ψ to Φ is defined by the formula

Ψ 𝑣 = max 𝑢 𝑣 − Φ 𝑢 : 𝑢 ≥ 0 .


An Orlicz function satisfies the condition Δ2, if there exists a constant 𝐾 > 0 such that

Φ 2𝑢 ≤ 𝐾Φ(𝑢)

for all 𝑢 ≥ 0 provided 𝜇 𝑇 = ∞, and for all 𝑢 large enough, provided 𝜇 𝑇 < ∞.

If the Orlicz function Φ satisfies the condition Δ2, then Köthe dual space is given by the formula

𝐿Φ,𝑝∗= 𝐿Ψ,𝑞

where 1

𝑝+

1

𝑞= 1 and Ψ is the complementary Orlicz

function to Φ.


An Orlicz space is reflexive if and only if both Orlicz functions: Φ and its complementary Ψ satisfy the appropriate (against the measure) condition Δ2.


The Δ2 − condition implies many good geometrical properties of the Orlicz space

𝐿Φ,𝑝.

In particular, the Δ2 − condition is sufficient

for the extreme points of the unit ball 𝐵(𝐿Φ,∞) to be closed.

But it is not sufficient.


An Orlicz function Φ is said to satisfy the Γ𝑎-condition if there exist constants 𝑐, 𝐾 > 1 and 𝑑 ≥ 0 such that 𝑑𝜇(𝑇) < ∞ and

Φ 𝑡 ≤ 𝐾 ⋅ Φ (𝑠) + 𝑑

for every 𝑡 ∈ 𝑆𝐶 Φ ∩ [0, 𝑐𝑠] and 𝑠 ∈ 𝑆𝐶(Φ).

A.Suarez-Granero, MW

The set 𝑒𝑥𝑡 𝐵(𝐿Φ,∞) is closed if and only if the Orlicz function Φ satisfies the Γ𝑎-condition.


The problem of characterization the closedness of the set of extreme points of the unit ball of Orlicz spaces equipped with the Orlicz norm (𝑝 = 1) or the p-Amemiya norm (1 < 𝑝 < ∞) is far more complicated.

It occurs that the condition Δ2 is not important in that case. The main role plays the set 𝑆𝐶Φ of all points of strict convexity of the graph of the function Φ.


Define:

𝑏Φ∗ = inf 𝑢 ≥ 0:Ψ 𝑝+ 𝑢 𝜇 𝑇 ≥ 1

𝑏Φ∗∗ = sup 𝑢 ≥ 0:Ψ 𝑝+ 𝑢 𝜇 𝑇 ≤ 1

Theorem

Let Φ be an Orlicz function such that sup 𝑆𝐶Φ = ∞.

Then the set 𝑒𝑥𝑡 𝐵(𝐿Φ,1) is closed if and only if one of the following conditions is satisfied:

◦ (i) 𝑏Φ∗∗ = ∞ (i.e., the Orlicz space 𝐿Φ,1 is linearly isometric to

the Lebesgue space 𝐿1),

◦ (ii) Φ is strictly convex on the interval (𝑏Φ∗ , ∞) and Φ does

not admit an asymptote at infinity.


For any Köthe space 𝐸 and any Orlicz function Φ, on the space 𝐿0 of 𝜇-measurable functions we

define the convex semimodular 𝜌𝜑 by the formula

𝜌Φ 𝑥 = Φ ∘ 𝑥 𝐸 if Φ ∘ 𝑥 ∈ 𝐸

𝜌Φ 𝑥 = ∞, otherwise.

The Calderon-Lozanovskii space 𝐸Φ generated by the couple (𝐸,Φ) is defined as the set

𝐸Φ = { 𝑥 ∈ 𝐿0 ∶ ∃𝜆 > 0 𝜌Φ(𝜆𝑥) < ∞ }.

In the Calderon-Lozanovskii space 𝐸Φ we define a

norm by the formula 𝑥 Φ = inf{ 𝜆 > 0 ∶ 𝜌Φ𝑥

𝜆≤ 1}.


By a Köthe space 𝐸 ⊂ 𝐿0 we mean a Banach space (𝐸, || ⋅ ||𝐸) satisfying the following conditions:

(𝑖) for every 𝑥 ∈ 𝐿0 and 𝑦 ∈ 𝐸 such that 𝑥 𝑡 ≤ |𝑦(𝑡)| for 𝜇-a.e. 𝑡 we have 𝑥 ∈ 𝐸 and 𝒙 𝑬 ≤ 𝒚 𝑬,

(𝑖𝑖) there is a function 𝑥 ∈ 𝐸 such that 𝑥(𝑡) > 0 for 𝜇-a.e. 𝑡.


If 𝐸 = 𝐿1 then the Calderon-Lozanovskii space 𝐸Φ = 𝐿1 Φ coincides with the Orlicz space 𝐿Φ,∞.

Question: What is the relation between closedness of the sets 𝑒𝑥𝑡 𝐵(𝐸) and 𝑒𝑥𝑡 𝐵(𝐸Φ)?


(𝑁𝜇)-property: ◦ For every sequence (𝑥𝑛) in 𝑆(𝐸Φ) and 𝑥 ∈ 𝑆(𝐸Φ),

𝑥𝑛 − 𝑥 Φ → 0 ⇒ Φ ∘ 𝑥𝑛 𝜇→ Φ ∘ 𝑥 .

Example

If the Köthe space 𝐸 is symmetric then the norm convergence in 𝐸 implies the convergence in the measure 𝜇 , whence 𝐸Φ satisfies the (𝑁𝜇)-property as well (since 𝐸Φ is symmetric in that case).


Condition (𝑒):

For every point 𝑥 ∈ 𝑆(𝐸Φ) 𝑥 ∈ 𝑒𝑥𝑡 𝐵 𝐸Φ ⇒ Φ ∘ 𝑥 ∈ 𝑒𝑥𝑡 𝐵(𝐸).

Example:

Let Φ(𝑢) = 𝑚𝑎𝑥{0, |𝑢| − 1} for 𝑢 ≥ 0.

For every Köthe space 𝐸 with e𝑥𝑡 𝐵 𝐸 ≠ ∅ the space 𝐸Φ satisfies the condition (𝑒).


(𝐻𝜇) - Kadec-Klee property with respect to the convergence in measure:

A Köthe space 𝐸 has the (𝐻𝜇)-property if for an arbitrary sequence (𝑥𝑛) in 𝑆(𝐸) and an arbitrary 𝑥 ∈ 𝑆(𝐸) we have

𝑥𝑛 𝜇→ 𝑥 ⇒ 𝑥𝑛 − 𝑥 𝐸 → 0.


A point 𝑥 ∈ 𝐸+ is called a point of upper monotonicity (𝑈𝑀-point) if for any 𝑦 ∈ 𝐸+\{0} we have 𝑥 𝐸 < 𝑥 + 𝑦 𝐸.

If every point of 𝑆(𝐸+) is a 𝑈𝑀-point then the space 𝐸 is strictly monotone.

The relation between 𝑈𝑀-points and extreme points in Köthe space reads as follows:

Let 𝐸 be an arbitrary Köthe space. A point 𝑥 ∈ 𝑆(𝐸) is an extreme point of 𝐵(𝐸) if and only if |𝑥| is an 𝑈𝑀-point and 𝑥 ∈ 𝑒𝑥𝑡 𝐵(𝐸+).


Let 𝐸Φ be a Calderon-Lozanovskii space with the properties (𝑁𝜇) and (𝑒). Moreover, assume that 𝐸 is

a Köthe space with the (𝐻𝜇)-property and the set of 𝑈𝑀-points of 𝑆(𝐸+) is closed.

If Φ is a strictly convex function with Φ ∈ Δ2𝐸, then

the set 𝑒𝑥𝑡 𝐵(𝐸Φ) is closed if and only if the set 𝑒𝑥𝑡 𝐵(𝐸) is closed.


Thank you for your attention!

25 Positivity VII, Leiden, July 22-26, 2013, The Netherlands

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