On the class of weak almost limited operators

Abstract

We introduce and study the class of weak almost limited operators. We establish a characterization of pairs of Banach lattices E, F for which every positive weak almost limited operator T : E→F is almost limited (resp. almost Dunford- Pettis). As consequences, we will give some interesting results.     

A NON-LINEAR ℓ 1 PROBLEM

Abstract

In this note we consider a non-linear problem posed to the author in a private communication with Per Enflo: If a normed space X contains a non-linear ? ₁ ⁽ⁿ⁾-cube (where n ≥ 2 is a natural number) does it necessarily contain a linear isometric copy of ? ₁ ⁽ⁿ⁾?

We exhibit a strong regularity property of non-linear ? ₁ ⁽ⁿ⁾ cubes and apply it to obtain an affirmative answer to Enflo's problem in the setting X = l _∞ ^(m) that, moreover, coincides precisely with well known linear theory.     

Isometries between the unit spheres of lβγ -sum of strictly convex normed spaces

Abstract

In this paper, the author introduces a new F-space the l ^βγ-sum of strictly convex normed spaces, and obtains the representation of onto isometry between the unit spheres, then concludes that such mappings can be extended to the whole space as real linear isometries.     

PROJECTIONS ON OPERATOR DUALS

Abstract

We discuss the existence of a projection with kernel K_b(E,F) ¹ (the annihilator of the quasi-compact operators) on the dual space of the space L _b,(E, F) of continous linear operators. Our results are proved in the context of Hausdorff locally convex spaces, but also provide extensions of recent results in the context of Banach spaces.     

A SUPPLEMENT TO MY PAPER ON REAL AND COMPLEX OPERATOR IDEALS

Abstract

We show that the main problem left open in [2] can be solved using the Banach spaces Z^α recently constructed by Kalton [1]. This gives an example of a complex operator ideal that has no real analogue. It thus shows the richer structure of complex operator ideals compared with the real ones.     

Dunford-Pettis like properties on tensor products

In this paper we study equivalent formulations of the DP^? P_p (1 < p < ∞). We show that X has the DP^? P_p if and only if every weakly-p-Cauchy sequence in X is a limited subset of X. We give su?cient conditions on Banach spaces X and Y so that the projective tensor product X ?_π Y, the dual (X ?_? Y)^? of their injective tensor product, and the bidual (X ?_π Y)^?? of their projective tensor product, do not have the DP P_p, 1 < p < ∞. We also show that in some cases, the projective and the injective tensor products of two spaces do not have the DP^? P_p, 1 < p < ∞.     

Schauder type theorems for differentiable and holomorphic mappings

Denoting byC _wu ^p (E) the algebra of allC ^p-real-valued functions on the real Banach spaceE such that the functions and the derivatives are weakly uniformly continuous on bounded subsets, it is known that the algebra homomorphismsA:C _wu ^q (F)C _wu ^p (E) are induced by differentiable mappingsg:EF ^**. We prove that, for 1p+1q, the following are equivalent: (a)A is compact; (b)g and its derivatives are compact; (c)gC _wu ^p (E,F ^**) (the authors had proved that, forp=q<,A is [weakly] compact if and only ifg is a constant mapping, and it is known that ifq<p, thenA is always induced by a constant mapping and is therefore compact). Also, for an entire function of bounded typegH _b (U,F), where is a balanced open subset, andE,F are complex Banach spaces, lettingA:H _b (F)H _b (U) be the homomorphism given byA(f)=fg for allfH _b (F), we prove thatA is compact if and only ifg is compact.Supported in part by DGICYT Grant PB 94-1052 (Spain).Supported in part by DGICYT Grant PB 93-0452 (Spain).     

Gelfand Numbers of Identity Operators Between Symmetric Sequence Spaces

Complementing and generalizing classical as well as recent results, we prove asymptotically optimal formulas for the Gelfand and approximation numbers of identities Eⁿ ↪ Fⁿ, where Eⁿ and Fⁿ denote the n-th sections of symmetric quasi-Banach sequence spaces E and F satisfying certain interpolation assumptions. We illustrate our results by considering classical spaces such as Lorentz and Orlicz sequence spaces. Supported by DFG grant Hi 584/2-2.     

Zero sets of polynomials in several variables

Let , where n is odd. We show that there is an integer N = N(k, n) such that for every n-homogeneous polynomial there exists a linear subspace , such that P|_X ≡ 0. This quantitative estimate improves on previous work of Birch et al., who studied this problem from an algebraic viewpoint. The topological method of proof presented here also allows us to obtain a partial solution to the Gromov-Milman problem (in dimension two) on an isometric version of a theorem of Dvoretzky. Received: 28 September 2004     

On extensions of Pełczyński’s decomposition method in Banach spaces

Let X and Y be Banach spaces such that each of them is isomorphic to a complemented subspace of the other. In 1996, W. T. Gowers solved the Schroeder-Bernstein Problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In this paper, we give suitable conditions on finite sums of X and Y to yield that X^m is isomorphic to Yⁿ for some In other words, we obtain some extensions of the well-known Pełczyński decomposition method in Banach spaces. In order to do this, we introduce the notion of Nearly Schroeder-Bernstein Quadruples for Banach spaces and pose a Conjecture to characterise them. Received: 5 January 2005     

Decomposition Methods in Banach Spaces via Supplemented Subspaces Resembling Pełczyński’s Decomposition Methods

Suppose that X and Y are Banach spaces complemented in each other with supplemented subspaces A and B. In 1996, W. T. Gowers solved the Schroeder–Bernstein problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In this paper, we obtain some suitable conditions involving the spaces A and B to yield that X is isomorphic to Y or to provide that at least X ^m is isomorphic to Yⁿ for some m, n ∈ IN^*. So we get some decomposition methods in Banach spaces via supplemented subspaces resembling Pełczyński’s decomposition methods. In order to do this, we introduce several notions of Schroeder–Bernstein Quadruples acting on the spaces X, Y, A and B. Thus, we characterize them by using some Banach spaces recently constructed. Received: October 4, 2005.     

Applications of a decomposition inX⊗λ Y

A Schauder Decomposition inX⊗_λ when Y has a Schauder Decomposition is presented. This decomposition is then used to investigate the Approximation Property. For instance, it is shown that if X has the Approximation Property and Y has a basis thenX⊗_λ has the Approximation Property. We also use this to obtain results concerning operators on vector valued continuous function spaces. Entrata in Redazione il 31 marzo 1999     

Operators of Fourier type <Emphasis Type="Italic">p</Emphasis> with respect to some subgroups of a locally compact Abelian group

It is shown that all Banach space operators which have Fourier type p(1 < p 2) with respect to a second countable locally compact abelian group G also have Fourier type p with respect to every closed discrete subgroup H of G. The same statement holds for any closed subgroup H of G when p = 2. Also shown as a corollary is that Fourier type 2 operators have Walsh type 2. Received: 10 June 2002     

A Dodds–Fremlin Property for Asplund and Radon–Nikodým Operators

Let E and F be Banach lattices and let S, T: E→ F be positive operators such that 0≤ S≤ T. It is shown that if T is a Radon–Nikodym operator, F has order continuous norm and E and F both have (Schaefer's) property (P), then S is a Radon–Nikodym operator; also, if T is an Asplund operator, E' has order continuous norm and E has property (P), then S is an Asplund operator.     

Ideal properties of the Dunford integration operator

Let F be a Banach space. We establish necessary and sufficient conditions for the Dunford integration operator, from the space of F‐valued Dunford integrable functions to the bidual of F, to belong to a given operator ideal. We also show how this fact can be used to characterize important classes of Banach spaces, such as Banach spaces with the Banach‐Saks property, separable Banach spaces not containing c₀, Banach spaces not containing c₀ or ?₁ and Asplund spaces not containing c₀.     

The distortion of Hilbert space

The unit sphere of Hilbert space, ₂, is shown to contain a remarkable sequence of nearly orthogonal sets. Precisely, there exist a sequence of sets of norm one elements of ₂, (C _i ) _i=1 , and reals _i 0 so that a) each setC _i has nonempty intersection with every infinite dimensional closed subspace of ₂ and b) forij,xC, andyC _j , |x, y|<_{min(i, j)} E. Odell was partially supported by NSF and TARP. Th. Schlumprecht was partially supported by NSF and LEQSF.     

Algebras of holomorphic mappings in (DF)-spaces

Let U be an absolutely convex open subset of a complex barrelled (DF)-space E and let F be a commutative Banach algebra with identity. Let H_b(U, F) be the space of holomorphic mappings from U into F that are bounded on the U-bounded sets and let H_b (U, F) be the space of the holomorphic mappings from U into F that are uniformly weakly continuous on the U-bounded sets, both endowed with the topology τ_b of uniform convergence on the U-bounded sets. The spectra of (H_wu (U, F), τ_b) and (H_b(U, F), τ_b) are studied.