Algebra descent spectrum of operators

We say that a Banach space X satisfies the “descent spectrum equality” (in short, DSE) whenever, for every bounded linear operator T on X, the descent spectrum of T as an operator coincides with the descent spectrum of T as an element of the algebra of all bounded linear operators on X. We prove that the DSE is fulfilled by ℓ₁, all Hilbert spaces, and all Banach spaces which are not isomorphic to any of their proper quotients (so, in particular, by the hereditarily indecomposable Banach spaces [8]), but not by ℓ _p , for 1 < p ≤ ∞ with p ≠ 2. Actually, a Banach space is not isomorphic to any of its proper quotients if and only if it is not isomorphic to any of its proper complemented subspaces and satisfies the DSE.     

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.     

Orthogonally Complemented Subspaces in Banach Spaces

In this paper, we extend the Moreau (Riesz) decomposition theorem from Hilbert spaces to Banach spaces. Criteria for a closed subspace to be (strongly) orthogonally complemented in a Banach space are given. We prove that every closed subspace of a Banach space X with dim X ≥ 3 (dim X ≤ 2) is strongly orthognally complemented if and only if the Banach space X is isometric to a Hilbert space (resp. strictly convex), which is complementary to the well-known result saying that every closed subspace of a Banach space X is topologically complemented if and only if the Banach space X is isomorphic to a Hilbert space.     

Embedding unconditional stable banach spaces into symmetric stable banach spaces

In this paper we prove the following result which solves a question raised by A. Pelczynski: “Every stable Banach space with an unconditional basis is isomorphic to a complemented subspace of some stable Banach space with a symmetric basis.” Moreover, we show that all the interpolation spacesl ^p ,l ^q _θ,X,1 1≦p, q<∞ andX stable, are stable.     

On symmetric basic sequences in lorentz sequence spaces II

It is shown that if {y _n} is a block of type I of a symmetric basis {x _n} in a Banach spaceX, then {y _n} is equivalent to {x _n} if and only if the closed linear span [y _n] of {y _n} is complemented inX. The result is used to study the symmetric basic sequences of the dual space of a Lorentz sequence spaced(a, p). Let {x _n,f _n} be the unit vector basis ofd(a, p), for 1≤p<+∞. It is shown that every infinite-dimensional subspace ofd(a, p) (respectively, [f _n] has a complemented subspace isomorphic tol _p (respectively,l _q, 1/p+1/q=1 when 1<p<+∞ andc ₀ whenp=1) and numerous other results on complemented subspaces ofd(a, p) and [f _n] are obtained. We also obtain necessary and sufficient conditions such that [f _n] have exactly two non-equivalent symmetric basic sequences. Finally, we exhibit a Banach spaceX with symmetric basis {x _n} such that every symmetric block basic sequence of {x _n} spans a complemented subspace inX butX is not isomorphic to eitherc ₀ orl _p, 1≤p<+∞.     

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.     

New examples of <Emphasis Type="Italic">c</Emphasis><Subscript>0</Subscript>-saturated Banach spaces

For every p ∈ (1, ∞), an isomorphically polyhedral Banach space E _p is constructed which has an unconditional basis and does not embed isomorphically into a C(K) space for any countable and compact metric space K. Moreover, E _p admits a quotient isomorphic to ℓ _p .     

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.     

Extension of multilinear maps defined on subspaces

We study the problem of whether every multilinear form defined on the product of n closed subspaces has an extension defined on the product of the entire Banach spaces. We prove that the property derived from this condition (the Multilinear Extension Property) is local. We use this to prove that, for a wide variety of Banach spaces, there exist a product of closed subspaces and a multilinear form defined on it, which has no extension to the product of the entire spaces. We show that the ℓ _p spaces, with 1 ≤p ≤ ∞ and p ≠ 2, are among them and, more generally, every Banach space which fails to have type p for some p < 2 or cotype q for some q > 2.     

Exact and explicit analytic solutions of an extended Jabotinsky functional differential equation

A Banach space X will be called extensible if every operator E → X from a subspace E ⊂ X can be extended to an operator X → X. Denote by dens X. The smallest cardinal of a subset of X whose linear span is dense in X, the space X will be called automorphic when for every subspace E ⊂ X every into isomorphism T: E → X for which dens X/E = dens X/TE can be extended to an automorphism X → X. Lindenstrauss and Rosenthal proved that c ₀ is automorphic and conjectured that c ₀ and ℓ₂ are the only separable automorphic spaces. Moreover, they ask about the extensible or automorphic character of c ₀(Γ), for Γ uncountable. That c ₀(Γ) is extensible was proved by Johnson and Zippin, and we prove here that it is automorphic and that, moreover, every automorphic space is extensible while the converse fails. We then study the local structure of extensible spaces, showing in particular that an infinite dimensional extensible space cannot contain uniformly complemented copies of ℓ _n ^p , 1 ≤ p < ∞, p ≠ 2. We derive that infinite dimensional spaces such as L _p (μ), p ≠ 2, C(K) spaces not isomorphic to c ₀ for K metric compact, subspaces of c ₀ which are not isomorphic to c ₀, the Gurarij space, Tsirelson spaces or the Argyros-Deliyanni HI space cannot be automorphic. The work of the first author has been supported in part by project MTM2004-02635     

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.     

On solutions to the Schroeder-Bernstein problem for Banach spaces

We investigate the geometry of the Banach spaces failing Schroeder-Bernstein Property (SBP). Initially we prove that every complex hereditarily indecomposable Banach space H is isomorphic to a complemented subspace of a Banach space S(H) that fails SBP in such a way that the only complemented hereditarily indecomposable subspaces of S(H) are those which are nearly isomorphic to H. Then we show that every Banach space having Mazur property is isomorphic to some complemented subspace of a Banach space which is not isomorphic to its square but isomorphic to its cube. Finally, we prove that if a Banach space X fails SBP then either it is not primary or the Grothendieck group K₀(L(X)) of the algebra of operators on X is not trivial.     

Embeddings ofC(Δ) andL 1[0, 1] in Banach lattices

It is proved that ifE is a separable Banach lattice withE′ weakly sequentially complete,F is a Banach space andT:E→F is a bounded linear operator withT′F′ non-separable, then there is a subspaceG ofE, isomorphic toC(Δ), such thatT _G is an isomorphism, whereC(Δ) denotes the space of continuous real valued functions on the Cantor discontinuum. This generalizes an earlier result of the second-named author. A number of conditions are proved equivalent for a Banach latticeE to contain a subspace order isomorphic toC(Δ). Among them are the following:L ¹ is lattice isomorphic to a sublattice ofE′;C(Δ)′ is lattice isomorphic to a sublattice ofE′; E contains an order bounded sequence with no weak Cauchy subsequence;E has a separable closed sublatticeF such thatF′ does not have a weak order unit. The research of both authors was partially supported by the National Science Foundation, NSF Grant No MPS 71-02839 A04.     

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.     

On factors ofC([0, 1]) with non-separable dual

LetC denote the Banach space of scalar-valued continuous functions defined on the closed unit interval. It is proved that ifX is a Banach space andT:C→X is a bounded linear operator withT ^* X ^* non-separable, then there is a subspaceY ofC, isometric toC, such thatT|Y is an isomorphism. An immediate consequence of this and a result of A. Pelczynski, is that every complemented subspace ofC with non-separable dual is isomorphic (linearly homeomorphic) toC. The research for this paper was partially supported by NSF-GP-30798X. An erratum to this article is available at .     

On Subspaces and Quotients of Banach Spaces C ( K , X )

 We study the relation of to the subspaces and quotients of Banach spaces of continuous vector-valued functions , where K is an arbitrary dispersed compact set. More precisely, we prove that every infinite dimensional closed subspace of totally incomparable to X contains a copy of complemented in . This is a natural continuation of results of Cembranos-Freniche and Lotz-Peck-Porta. We also improve our result when K is homeomorphic to an interval of ordinals. Next we show that complemented subspaces (resp., quotients) of which contain no copy of are isomorphic to complemented subspaces (resp., quotients) of some finite sum of X. As a consequence, we prove that every infinite dimensional quotient of which is quotient incomparable to X, contains a complemented copy of . Finally we present some more geometric properties of spaces.     

On Subspaces and Quotients of Banach Spaces C(K, X)

 We study the relation of to the subspaces and quotients of Banach spaces of continuous vector-valued functions , where K is an arbitrary dispersed compact set. More precisely, we prove that every infinite dimensional closed subspace of totally incomparable to X contains a copy of complemented in . This is a natural continuation of results of Cembranos-Freniche and Lotz-Peck-Porta. We also improve our result when K is homeomorphic to an interval of ordinals. Next we show that complemented subspaces (resp., quotients) of which contain no copy of are isomorphic to complemented subspaces (resp., quotients) of some finite sum of X. As a consequence, we prove that every infinite dimensional quotient of which is quotient incomparable to X, contains a complemented copy of . Finally we present some more geometric properties of spaces. Received 8 November 2000; in revised form 7 December 2001     

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.     

On Tsirelson’s space

A structure theory is developed for Tsirelson’s example of a Banach space which contains no isomorphic copy ofl _p orc ₀. In particular, it is shown that this space is the first example, other than subspaces ofl _p andc ₀, of a Banach space which embeds isomorphically into each of its infinite dimensional subspaces. Supported in part by NSF MCS-8002221 and MCS-8102238. Supported in part by NSF MCS-7903042.     

On perfectly homogeneous bases in Banach spaces

It is proved that a Banach space is isomorphic toc _o or tol _p if and only if it has a normalized basis {χ_i i _{} i=1} ^∞ which is equivalent to every normalized block-basis with respect to {χ_i i _{} i=1} ^∞ . This is part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Prof. A. Dvoretzky and Dr. J. Lindenstrauss. The author wishes to thank Dr. Lindenstrauss for his helpful guidance and for the interest he showed in the paper, and the referee for his valuable remakrs.