Extending operators into Lindenstrauss spaces

Abstract

We study the global and local approaches to the problem of extension of operators into Lindenstrauss spaces.

Extension conditions for bounded linear and sublinear operators with values in lindenstrauss spaces

“…Refer to the survey [1] for the main results of the extension theory of bounded linear operators, the unsolved problems, and the bibliography. The recent article [2] continues studying linear operators with values in Lindenstrauss spaces and states new problems, one of which we solve here.…”

“…In the last few years interest arouses in the extension problem for both linear and nonlinear mappings, including Lipschitz, Hölder, z-linear, and bilinear mappings (see [2][3][4][5][6][7][8][9][10] for instance). Nevertheless, as far as the author is aware, the extension questions remain untouched for the class of continuous sublinear operators which is the nearest class to the class of continuous linear operators (see [11]).…”

“…It amounts to the existence of a weakly*-weakly* continuous mapping from B E * to λ · B X * , where B E * and B X * stand for the unit balls with center zero of some Banach spaces E * and X * . Recently [2,Lemma 2.2] for some classes of spaces, which are Lindenstrauss spaces, some partial criteria were obtained and then the problem is posed of finding a criterion for the entire class of Lindenstrauss spaces [2, p. 12]. A key to solving this problem as well as other questions is the author's theorem on a representation of sublinear operators with values in L by multivalued mappings [13,Theorem 1].…”

The structure of Lindenstrauss–Pełczyński spaces

“…The spaces are so named because Lindenstrauss and Pełczyński first proved in [9] that C(K)-spaces have this property. In [6] it was shown that every LP-space is an L ∞ -space, that not all L ∞ -spaces are LP-spaces, and that complemented subspaces of Lindenstrauss spaces (see also [9,7]), separably injective spaces and L ∞ -spaces not containing c 0 are LP-spaces.We now prove a fundamental structure theorem for this class; namely, separable LP-spaces are characterized as those L ∞ Banach spaces having all subspaces of c 0 placed in a unique position. Precisely, let Y, X be Banach spaces.…”

“…The spaces are so named because Lindenstrauss and Pełczyński first proved in [9] that C(K)-spaces have this property. In [6] it was shown that every LP-space is an L ∞ -space, that not all L ∞ -spaces are LP-spaces, and that complemented subspaces of Lindenstrauss spaces (see also [9,7]), separably injective spaces and L ∞ -spaces not containing c 0 are LP-spaces.…”