prof. marek wisła adama mickiewicz university in poznań, poland positivity vii, zaanen centennial...
TRANSCRIPT
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
2
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
3
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.
Positivity VII, Leiden, July 22-26, 2013, The Netherlands
Nice operators
4
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
Positivity VII, Leiden, July 22-26, 2013, The Netherlands
Finite dimensional spaces?
5
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
Positivity VII, Leiden, July 22-26, 2013, The Netherlands
6
Characterize those Banach spaces in which the set of extreme points of the unit ball is closed.
Samples : OK, since . OK.
Goal
Positivity VII, Leiden, July 22-26, 2013, The Netherlands
7
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
Positivity VII, Leiden, July 22-26, 2013, The Netherlands
8
Examples of Orlicz functions
Positivity VII, Leiden, July 22-26, 2013, The Netherlands
9
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
Positivity VII, Leiden, July 22-26, 2013, The Netherlands
10
If is an Orlicz function, then the complementary function to is defined by the formula
.
Complementary function
Positivity VII, Leiden, July 22-26, 2013, The Netherlands
11
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
Positivity VII, Leiden, July 22-26, 2013, The Netherlands
12
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
Positivity VII, Leiden, July 22-26, 2013, The Netherlands
13
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.
Positivity VII, Leiden, July 22-26, 2013, The Netherlands
Closedness of
14
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.
Positivity VII, Leiden, July 22-26, 2013, The Netherlands
Closedness of
15
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 .
Positivity VII, Leiden, July 22-26, 2013, The Netherlands
Closedness of ,
16
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.
Positivity VII, Leiden, July 22-26, 2013, The Netherlands
Closedness of
17
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 .
Positivity VII, Leiden, July 22-26, 2013, The Netherlands
Calderon-Lozanovskii space
18
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. .
Positivity VII, Leiden, July 22-26, 2013, The Netherlands
Köthe space
19
If then the Calderon-Lozanovskii spacecoincides with the Orlicz space .
Question: What is the relation between closedness of the sets and ?
Positivity VII, Leiden, July 22-26, 2013, The Netherlands
Question
20
-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
Positivity VII, Leiden, July 22-26, 2013, The Netherlands
21
Condition : For every point
Example:Let for . For every Köthe space with the space satisfies the condition .
Condition
Positivity VII, Leiden, July 22-26, 2013, The Netherlands
22
- 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
Positivity VII, Leiden, July 22-26, 2013, The Netherlands
23
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
Positivity VII, Leiden, July 22-26, 2013, The Netherlands
24
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
Positivity VII, Leiden, July 22-26, 2013, The Netherlands
25
Thank you for your attention!
Positivity VII, Leiden, July 22-26, 2013, The Netherlands