Prof. Marek Wisła Adama Mickiewicz University in Poznań, Poland

CLOSEDNESS OF THE SET OF EXTREME POINTS OF THE UNIT BALL IN ORLICZ AND CALDERON-LOZANOVSKII SPACES

Prof. Marek WisłaAdama 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 from a Banach space to another Banach sapace is called compact if the image under of any bounded subset of is a relatively compact subset of

Assume that is a compact Hausdorff space. To any linear operator we can associate a continuous function defined by the formula

, .

Compact operators

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

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A linear operator is called nice if

where denotes the set of extreme points of the unit ball of the Banach space

Blumenthal, Lindenstrauss, Phelps A compact linear operator from a Banach

space into the space of continuous functions is extreme provided it is nice.


Nice operators

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Blumenthal, Lindenstrauss, Phelps If is a finite dimensional normed linear space

such that the or the unit ball is plyhedron then is a dense subset of for every extreme linear operator .

B.L.P. gave an example of a four dimensional

Banach space and an extreme linear operator such that for every


Finite dimensional spaces?

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The nice condition can be weakened as long as the set of extreme points is closed, namely it suffices to assume than

for some dense subset . Indeed,

.

Almost nice operators


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

Samples : OK, since . OK.

Goal


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A function is called an Orlicz function, if, is not identically equal to 0, it is even, continuous and convex on the interval and left-continuous at, where .

We shall denote .

Orlicz function


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Examples of Orlicz functions


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By the Orlicz space we mean the space of all –integrable functions with a constant , i.e., for some .

By p-Amemiya norm we mean the functional defined by

, if , , if .

Orlicz space


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

.

Complementary function


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An Orlicz function satisfies the condition , if there exists a constant such that

for all provided , and for all large enough, provided .

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

where and is the complementary Orlicz function to .

Köthe dual space


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

Reflexive Orlicz spaces


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The condition implies many good geometrical properties of the Orlicz space .

In particular, the condition is sufficient for the extreme points of the unit ball to be closed.

But it is not sufficient.


Closedness of

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An Orlicz function is said to satisfy the -condition if there exist constants and such that and

for every and .

A.Suarez-Granero, MW The set is closed if and only ifsatisfies the-

condition.


Closedness of

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

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


Closedness of ,

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Define:

Theorem Let be an Orlicz function such that . Then the set

is closed if and only if one of the following conditions is satisfied:◦ (i) (i.e., the Orlicz space is linearly isometric to the

Lebesgue space ),◦ (ii) is strictly convex on the interval and does not admit

an asymptote at infinity.


Closedness of

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For any Köthe space and any Orlicz function , on the space of -measurable functions we define the convex semimodular by the formula

if , otherwise.

The Calderon-Lozanovskii space generated by the couple is defined as the set

. In the Calderon-Lozanovskii space we define a

norm by the formula .


Calderon-Lozanovskii space

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By a Köthe space we mean a Banach space satisfying the following conditions: for every and such that for -a.e. we have and , there is a function such that for -a.e. .


Köthe space

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If then the Calderon-Lozanovskii spacecoincides with the Orlicz space .

Question: What is the relation between closedness of the sets and ?


Question

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-property: ◦ For every sequence in and ,

Example If the Köthe space is symmetric then the

norm convergence in implies the convergence in the measure , whence satisfies the -property as well (since is symmetric in that case).

(𝑵𝝁 )−𝒑𝒓𝒐𝒑𝒆𝒓𝒕𝒚


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Condition : For every point

Example:Let for . For every Köthe space with the space satisfies the condition .

Condition


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

A Köthe space has the -property if for an arbitrary sequence in and an arbitrary we have

.

Kadec-Klee property


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A point is called a point of upper monotonicity (-point) if for any we have .

If every point of is a -point then the space is strictly monotone.

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

Let be an arbitrary Köthe space. A point is an extreme point of if and only if is an -point and .

-points


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Let be a Calderon-Lozanovskii space with the properties and . Moreover, assume that is a Köthe space with the -property and the set of -points of is closed. If is a strictly convex function with , then the set is closed if and only if the set is closed.

Closedness of


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Thank you for your attention!


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Prof. Marek Wisła Adama Mickiewicz University in Poznań, Poland

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