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Prof. Marek Wisła Adama Mickiewicz University in Poznań, Poland

CLOSEDNES S OF THE SET OF EXTREME POINTS OF THE UNIT BALL IN ORLICZ AND CALDERON-LOZANOVSKII SPACES. Photo: Sandra Sardjono. Prof. Marek Wisła Adama Mickiewicz University in Poznań, Poland Positivity VII, Zaanen Centennial Conference, Leiden July 22-26, 2013, The Netherlands.

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Prof. Marek Wisła Adama Mickiewicz University in Poznań, Poland

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  1. CLOSEDNESSOF THE SET OF EXTREME POINTS OF THE UNIT BALL IN ORLICZ AND CALDERON-LOZANOVSKII SPACES Photo: Sandra Sardjono Prof. Marek Wisła Adama Mickiewicz University in Poznań, Poland Positivity VII, Zaanen Centennial Conference, Leiden July 22-26, 2013, The Netherlands

  2. Compact operators • A linear operator from a Banach space to another Banachsapace 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 , . Positivity VII, Leiden, July 22-26, 2013, The Netherlands

  3. Nice operators • 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

  4. Finitedimensionalspaces? 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

  5. Almost nice operators The nice condition canbe weakened as long as the set of extreme points is closed, namely it suffices to assume than for some dense subset . Indeed, . Positivity VII, Leiden, July 22-26, 2013, The Netherlands

  6. Goal • Characterize those Banach spaces in which the set of extreme points of the unit ball is closed. • Samples • : OK, since . • OK. Positivity VII, Leiden, July 22-26, 2013, The Netherlands

  7. Orlicz function • 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 . Positivity VII, Leiden, July 22-26, 2013, The Netherlands

  8. Examples of Orlicz functions Positivity VII, Leiden, July 22-26, 2013, The Netherlands

  9. Orlicz space • 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 . Positivity VII, Leiden, July 22-26, 2013, The Netherlands

  10. Complementary function • If is an Orlicz function, then the complementary function to is defined by the formula . Positivity VII, Leiden, July 22-26, 2013, The Netherlands

  11. Köthe dual space • An Orliczfunction satisfies the condition , if there exists a constantsuch that for all provided , and for all large enough, provided . • If theOrliczfunctionsatisfies the condition , thenKöthedual spaceisgiven by the formulawhere and is the complementary Orlicz function to . Positivity VII, Leiden, July 22-26, 2013, The Netherlands

  12. Reflexive Orlicz spaces • An Orlicz space is reflexive if and only if both Orlicz functions: and its complementary satisfy the appropriate (against the measure) condition . Positivity VII, Leiden, July 22-26, 2013, The Netherlands

  13. Closedness of • 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

  14. Closedness of • 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

  15. Closedness of , • 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

  16. Closedness of 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

  17. Calderon-Lozanovskiispace • For any Köthespace and any Orlicz function , on the space of -measurable functions we define the convex semimodularby the formula if , otherwise. • The Calderon-Lozanovskii space generated by the couple is defined as the set . • In the Calderon-Lozanovskiispace we define a norm by the formula . Positivity VII, Leiden, July 22-26, 2013, The Netherlands

  18. Köthespace • By a Köthespace 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

  19. Question • 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

  20. -property: • For every sequence in and , • Example • If the Köthespace 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). Positivity VII, Leiden, July 22-26, 2013, The Netherlands

  21. Condition • Condition : For every point Example: Let for . For every Köthespace with the spacesatisfies the condition . Positivity VII, Leiden, July 22-26, 2013, The Netherlands

  22. Kadec-Kleeproperty • - Kadec-Klee property with respect to the convergence in measure: • A Köthespace has the -property if for an arbitrary sequence in and an arbitrary we have . Positivity VII, Leiden, July 22-26, 2013, The Netherlands

  23. -points • A point is called a point of upper monotonicity(-point) if for any wehave . • If every point of is a -pointthen the space isstrictly monotone. • The relation between -points and extreme points in Köthespace reads as follows: • Let be an arbitrary Köthespace. A point is an extreme point of if andonly if is an -point and . Positivity VII, Leiden, July 22-26, 2013, The Netherlands

  24. Closedness of • Let be a Calderon-Lozanovskii space with the properties and .Moreover, assume that is a Köthespace 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. Positivity VII, Leiden, July 22-26, 2013, The Netherlands

  25. Thank you for your attention! Positivity VII, Leiden, July 22-26, 2013, The Netherlands

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