Positivity (2013) 17:223–234
Geometrical properties of Banach spaces generated
by sublinear operators
S. V. Astashkin
Received: 22 July 2011 / Accepted: 27 January 2012 / Published online: 12 February 2012
© Springer Basel AG 2012
Abstract We solve a problem posed by Mastylo (Math Japon 36(1), 85–92, 1991)
proving that every “non-trivial” subspace of a Banach space X generated by some
positive sublinear operator and an L
-space with 1 ≤ p < ∞ contains, for any
ε>0, an (1 + ε)-copy of l
which is (1 + ε)-complemented in X.
Keywords Sublinear operator · Complemented l
-copy · Real interpolation space ·
Extrapolation space · Cesàro space
Mathematics Subject Classiﬁcation (2000) Primary 46B20 · 46E30;
Secondary 46B42 · 46B70
1 Introduction and preliminaries
Many Banach spaces which play an important role in functional analysis and its appli-
cations are obtained in a special way: the norms of these spaces are generated by
positive sublinear operators. The well-known examples of such spaces are real inter-
polation spaces (see [1,2]), extrapolation spaces (see [3–6]), Triebel spaces B
, “tent” spaces .
Mastylo  investigated under some conditions Banach spaces of the above type.
In this paper, we weaken in a natural way the conditions imposed there and give a
solution of a problem of existing complemented l
-copies in such spaces raised in .
Research was partially supported by RFBR Grant no. 10-01-00077.
S. V. Astashkin (
Department of Mathematics and Mechanics, Samara State University,
Acad. Pavlov 1, 443011 Samara, Russia