On orlicz sequence spaces

It is proved that every Orlicz sequence space contains a subspace isomorphic to somel _p . The question of uniqueness of symmetric bases in Orlicz sequence spaces is investigated.     

On Orlicz sequence spaces. II

Given a separable Orlicz sequence spacel _F we investigate those Orlicz sequence spacesl _f which are isomorphic to subspaces (respectively complemented subspaces) ofl _F. We give in particular an example of a reflexive Orlicz sequence space which does not contain anyl _p, 1<p<∞, as a complemented subspace.     

Über absolut repräsentierende Systeme aus Quasipolynomen in Räumen analytischer Funktionen

Criteria for Orlicz spaces containing an isomorphic as well as an almost isometric complemented copy of l¹ are given. The same is made for copies of l^∞ and c₀ in Orlicz spaces and in the subspace of order continuous elements in Orlicz space, respectively. The Hahn-Banach theorem is not used in the constructions of the projections.     

On the subspaces ofL p (p>2) spanned by sequences of independent random variables

Let 2<p<∞. The Banach space spanned by a sequence of independent random variables inL ^p , each of mean zero, is shown to be isomorphic tol ²,l ^p ,l ²⊕l ^p , or a new spaceX _p , and the linear topological properties ofX _p are investigated. It is proved thatX _p is isomorphic to a complemented subspace ofL ^p and another uncomplemented subspace ofL ^p , whence there exists an uncomplemented subspace ofl ^p isomorphic tol ^p . It is also proved thatX _p is not isomorphic to the previously knownℒ _p spaces. The work for this research was partially supported by the National Science Foundation GP-12997.     

Finite representability ofl p(X) in Orlicz function spaces

We show that ifl _p(X),p ≠ 2, is finitely crudely representable in an Orlicz spaceL _ϕ (which does not containc ₀) then the Banach spaceX is isomorphic to a subspace ofL _p. The same remains true forp = 2 whenL _ϕ is 2-concave or 2-convex, or ifX has local unconditional structure. We extend a theorem of Guerre and Levy to Orlicz function spaces.     

On symmetric basic sequences in Lorentz sequence spaces

We examine the symmetric basic sequences in some classes of Banach spaces with symmetric bases. We show that the Lorentz sequence spaced(a,p) has a unique symmetric basis and every infinite dimensional subspace ofd(a,p) contains a subspace isomorphic tol ^p. The symmetric basic sequences ind(a,p) are identified and a necessary and sufficient condition for a Lorents sequence space with exactly two nonequivalent symmetric basic sequences in given. We conclude by exhibiting an example of a Lorentz sequence space having a subspace with symmetric basis which is not isomorphic either to a Lorentz sequence space or to anl ^p-space. This is part of the first author's Ph. D. thesis, prepared at the Hebrew University of Jerusalem under the supervision of Dr. L. Tzafriri.     

Null Sequences in Coechelon Spaces

Rotundity of finite -dimensional Orlicz spaces l^?_n equipped with the Luxemburg norm is considered. It is proved that criteria for rotundity of l^?_nfor n ≥ 3 does not depend on n and are the same as the criteria for rotundity of the inhite-dimensional subspace h^? of an Orlicz sequence spacel^?. Criteria for rotundity of l^?₂ are different. Next, criteria for exposed points, (H)- points, strongly exposed points and LUR- points of the unit sphere of l^? and of its subspace h^? are given.     

Orlicz function spaces without complemented copies ofl p

This paper proves the existence of Orlicz function spaces Lφ (0, 1) containing no complemented subspaces isomorphic tol ^p for anyp ≠ 2. Some properties of minimal Orlicz function spaces Lφ (0, 1) are also given. Supported in part by CAICYT grant 0338-84.     

Pitt's theorem for the Lorentz and Orlicz sequence spaces

LetL(X, Y) be the Banach space of all continuous linear operators fromX toY, and letK(X, Y) be the subspace of compact operators. Some versions of the classical Pitt theorem (ifp>q, thenK(l _p, l_q)=L(l_p, l_q)) for subspaces of Lorentz and Orlicz sequence spaces are established. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 18–25, January, 1997. Translated by V. N. Dubrovsky     

Banach spaces whose duals are isomorphic to l1(Γ)

Banach spaces X whose duals are isomorphic or isometric to l₁(Γ) are characterized by certain classes of operators on X. It is proved that a separable, conjugate space isomorphic to a complemented subspace of an L₁(S, Σ, μ) space is isomorphic to l₁; a $\text{L}$₁ space contained in a separable, conjugate space is isomorphic to a subspace of l₁.     

On the Orlicz function spacesL M (0, ∞)

The isomorphic properties of the Orlicz function spacesL _M (0, ∞) are investigated. Especially we treat the question, whether theL _p-spaces are the only symmetric function spaces on (0, ∞), which are isomorphic to a symmetric function space on (0, 1). For the class of slowly varying Orlicz functions we answer this in the affirmative, and we also prove some results concerning the general case, which indicate, that it might be true there also.     

Some isomorphically polyhedral orlicz sequence spaces

A Banach space is polyhedral if the unit ball of each of its finite dimensional subspaces is a polyhedron. It is known that a polyhedral Banach space has a separable dual and isc ₀-saturated, i.e., each closed infinite dimensional subspace contains an isomorph ofc ₀. In this paper, we show that the Orlicz sequence spaceh _M is isomorphic to a polyhedral Banach space if lim _t→0 M(Kt)/M(t)=∞ for someK<∞. We also construct an Orlicz sequence spaceh _M which isc ₀-saturated, but which is not isomorphic to any polyhedral Banach space. This shows that beingc ₀-saturated and having a separable dual are not sufficient for a Banach space to be isomorphic to a polyhedral Banach space.     

A Banach space with,up to equivalence,precisely two symmetric bases

It is well know that the classical sequence spaces c_o andl _p (1≦p<∞) have, up to equivalence, just one symmetric basis. On the other hand, there are examples of Orlicz sequence spaces which have uncountably many mutually non-equivalent symmetric bases. Thus in [4], p. 130, the question is asked whether there is a Banach space with, up to equivalence, more than one symmetric basis, but not uncountably many. In this paper we answer the question positively, by exhibiting a Banach space with, up to equivalence, precisely two symmetric bases.     

Uniform embeddings of metric spaces and of banach spaces into hilbert spaces

It is proved using positive definite functions that a normed spaceX is unifomly homeomorphic to a subset of a Hilbert space, if and only ifX is (linearly) isomorphic to a subspace of aL ₀(μ) space (=the space of the measurable functions on a probability space with convergence in probability). As a result we get thatl _p (respectivelyL _p (0, 1)), 2<p<∞, is not uniformly embedded in a bounded subset of itself. This answers negatively the question whether every infinite dimensional Banach space is uniformly homeomorphic to a bounded subset of itself. Positive definite functions are also used to characterize geometrical properties of Banach spaces. Partially supported by the National Science Foundation, Grant MCS-79-03322. Partially supported by the National Science Foundation, Grant MCS-80-06073.     

Universal non-compact operators between super-reflexive Banach spaces and the existence of a complemented copy of Hilbert space

Suppose that 1<p≦2, 2≦q<∞. The formal identity operatorI:l _p→l _qfactorizes through any given non-compact operator from ap-smooth Banach space into aq-convex Banach space. It follows that ifX is a 2-convex space andY is an infinite dimensional subspace ofX which is isomorphic to a Hilbert space, thenY contains an isomorphic copy ofl ₂ which is complemented inX.     

Complemented subspaces of (l 2⊕l 2⊕…) p (1<p<∞) with an unconditional basis

It is proved that every infinite dimensional complemented subspace of (l ₂⊕l ₂⊕…) _p (1<p<∞) with an unconditional basis is isomorphic to one of the following four spaces:l ₂,l _p,l ₂⊕l _p, (l ₂⊕l ₂⊕…) _p .     

Onl p-complemented copies in Orlicz spaces

Let 1<α≦β<∞ andF be an arbitrary closed subset of the interval [α,β]. An Orlicz sequence spacel ^φ (resp. an Orlicz function spaceL ^φ(μ)) with associated indices α and β is found in such a way that the set of valuesp for which thel ^p-space is isomorphic to a complemented subspace ofl ^φ (resp.L ^φ(μ)) is precisely the given setF (resp.F ∪ {2}). Also, a recent result of Hernández and Peirats [1] is extended showing that, even for the case in which the indices satisfy α_φ ^∞<2<β_φ ^∞, there exist minimal Orlicz function spacesL ^φ(μ) with no complemented copy ofl ^p for anyp ≠ 2. Supported in part by CAICYT grant 0338-84.     

Onl p-complemented copies in Orlicz spaces II

For anyp > 1, the existence is shown of Orlicz spacesL ^F andl ^F with indicesp containingsingular l ^p-complemented copies, extending a result of N. Kalton ([6]). Also the following is proved:Let 1 <α ≦β < ∞and H be an arbitrary closed subset of the interval [α, β].There exist Orlicz sequence spaces l ^F (resp. Orlicz function spaces L^F)with indices α and β containing only singular l ^p-complemented copies and such that the set of values p > 1for which l ^p is complementably embedded into l^F (resp. L ^F)is exactly the set H (resp. H ∪ {2&#x007D;). An explicitly defined class of minimal Orlicz spaces is given. Supported in part by CAICYT grant 0338-84.     

On the complemented subspaces ofX p

In this paper we prove some results related to the problem of isomorphically classifying the complemented subspaces ofX _p.We characterize the complemented subspaces ofX _pwhich are isomorphic toX _pby showing that such a space must contain a canonical complemented subspace isomorphic toX _p.We also give some characterizations of complemented subspaces ofX _pisomorphic tol _p⊕l ₂. Research supported in part by NSF grant DMS 890237.     

ℒ1 subspaces ofl 1

It is proved that every subspace ofl ₁ which is an Λ_{1, λ} space with λ close enough to 1 is isomorphic tol ₁.