The little Grothendieck theorem and Khintchine inequalities for symmetric spaces of measurable operators

We prove the little Grothendieck theorem for any 2-convex noncommutative symmetric space. Let M be a von Neumann algebra equipped with a normal faithful semifinite trace τ, and let E be an r.i. space on (0,∞). Let E(M) be the associated symmetric space of measurable operators. Then to any bounded linear map T from E(M) into a Hilbert space H corresponds a positive norm one functional f∈E(2)^∗(M) such that

     

Generalized degree for classes of finite dimensional upper hemi-continuous mappings in separable Banach spaces

Let E be a real separable Banach space, E^∗ the dual space of E, and Ω⊂E an open bounded subset, and let T:D(T)⊆E→2^E∗ be a finite dimensional upper hemi-continuous mapping with . A generalized degree theory is constructed for such a mapping. This degree is then applied to study the existence of approximate weak solutions to the equation 0∈Tx.     

Measures of weak noncompactness in Banach spaces

Measures of weak noncompactness are formulae that quantify different characterizations of weak compactness in Banach spaces: we deal here with De Blasi's measure ω and the measure of double limits γ inspired by Grothendieck's characterization of weak compactness. Moreover for bounded sets H of a Banach space E we consider the worst distance k(H) of the weak^∗-closure in the bidual of H to E and the worst distance ck(H) of the sets of weak^∗-cluster points in the bidual of sequences in H to E. We prove the inequalities

     

Metrizability of spaces of holomorphic functions

In this paper we prove that if U is an open subset of a metrizable locally convex space E of infinite dimension, the space H(U) of all holomorphic functions on U, endowed with the Nachbin-Coeuré topology τδ, is not metrizable. Our result can be applied to get that, for all usual topologies, H(U) is metrizable if and only if E has finite dimension.     

On an integral-type operator from logarithmic Bloch-type spaces to mixed-norm spaces on the unit ball

Let H(B) denote the space of all holomorphic functions on the open unit ball B of Cⁿ and g∈H(B). We characterize the boundedness and compactness of the following integral-type operator

     

The extension of the Krein-Šmulian theorem for Orlicz sequence spaces and convex sets

If X is a Banach space and C⊂X∗∗ a convex subset, for x∗∗∈X∗∗ and A⊂X∗∗ let be the distance from x∗∗ to C and . In this paper we prove that if φ is an Orlicz function, I an infinite set and X=?φ(I) the corresponding Orlicz space, equipped with either the Luxemburg or the Orlicz norm, then for every w^∗-compact subset K⊂X∗∗ we have if and only if φ satisfies the Δ₂-condition at 0. We also prove that for every Banach space X, every nonempty convex subset C⊂X and every w^∗-compact subset K⊂X∗∗ then and, if K∩C is w^∗-dense in K, then .     

Weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in Banach spaces

Let E be a real uniformly convex Banach space whose dual space E^∗ satisfies the Kadec-Klee property, K be a closed convex nonempty subset of E. Let be asymptotically nonexpansive mappings of K into E with sequences (respectively) satisfying kin→1 as n→∞, i=1,2,…,m, and . For arbitrary ?∈(0,1), let be a sequence in [?,1−?], for each i∈{1,2,…,m} (respectively). Let {xn} be a sequence generated for m?2 by

     

The approximation property for spaces of holomorphic functions on infinite-dimensional spaces I

For an open subset U of a locally convex space E, let (H(U),τ₀) denote the vector space of all holomorphic functions on U, with the compact-open topology. If E is a separable Fréchet space with the bounded approximation property, or if E is a (DFC)-space with the approximation property, we show that (H(U),τ₀) has the approximation property for every open subset U of E. These theorems extend classical results of Aron and Schottenloher. As applications of these approximation theorems we characterize the spectra of certain topological algebras of holomorphic mappings with values in a Banach algebra.     

Distances from selectors to spaces of Baire one functions

Given a metric space X and a Banach space (E,‖⋅‖) we study distances from the set of selectors Sel(F) of a set-valued map to the space B₁(X,E) of Baire one functions from X into E. For this we introduce the d-τ-semioscillation of a set-valued map with values in a topological space (Y,τ) also endowed with a metric d. Being more precise we obtain that

     

Integral-type operators acting between weighted-type spaces on the unit ball

Let B denote the unit ball in Cⁿ and H(B) the space of all holomorphic functions on B. We study the boundedness and compactness of the following integral-type operators

     

On affine difference sets and their multipliers

Let D be an affine difference set of order n in an abelian group G relative to a subgroup N. We denote by π(s) the set of primes dividing an integer and set H^∗=H?{ω}, where H=G/N and ω=∏_σ∈Hσ. In this article, using D we define a map g from H to N satisfying for iff {τ,τ⁻¹}={ρ,ρ⁻¹} and show that for any σ∈H^∗ and any integer m>0 with π(m)⊂π(n). This result is a generalization of J.C. Galati’s theorem on even order n [J.C. Galati, A group extensions approach to affine relative difference sets of even order, Discrete Mathematics 306 (2006) 42-51] and gives a new proof of a result of Arasu-Pott on the order of a multiplier modulo exp(H) ([K.T. Arasu, A. Pott, On quasi-regular collineation groups of projective planes, Designs Codes and Cryptography 1 (1991) 83-92] Section 5).     

Generalized composition operators from F(p, q, s) spaces to Bloch-type spaces

Let H(B) denote the space of all holomorphic functions on the unit ball B of Cⁿ. Let φ be a holomorphic self-map of B and g ∈ H(B) such that g(0) = 0. In this paper, we investigate the boundedness and compactness of the generalized composition operator

     

A Banach-Dieudonné theorem for germs of holomorphic functions

Let $\text{H}$(U) denote the space of all holomorphic functions on an open subset U of a complex Fréchet space E. Let $\text{H}$(K) denote the space of all holomorphic germs on a compact subset K of E. It is shown that $\text{H}$(K), with a natural topology, is the inductive limit of a suitable sequence of compact subsets, within the category of all topological spaces. As an application of this result it is shown that the compact-ported topology introduced by Nachbin coincides with the compact-open topology on $\text{H}$(U) whenever U is a balanced open subset of a Fréchet-Schwartz space. This last result improves earlier results of P. Boland and S. Dineen [Bull. Soc. Math. France106 (1978), 311–336], R. Meise [Proc. Roy. Irish Acad. Sect. A81 (1981), 217–223], and others.     

On an integral-type operator from iterated logarithmic Bloch spaces into Bloch-type spaces

Let H(B) denote the space of all holomorphic functions on the open unit ball B of Cⁿ. Let φ=(φ₁,…,φ_n) be a holomorphic self-map of B and g∈H(B) such that g(0)=0. In this paper we study the boundedness and compactness of the following integral-type operator, recently introduced by Xiangling Zhu and the second author

     

A characterization for ∗-isomorphisms in an indefinite inner product space

Let H₁ and H₂ be indefinite inner product spaces. Let L(H₁) and L(H₂) be the sets of all linear operators on H₁ and H₂, respectively. The following result is proved: If Φ is [∗]-isomorphism from L(H₁) onto L(H₂) then there exists such that Φ(T)=cUTU[∗] for all T∈L(H₁) with UU[∗]=cI₂, U[∗]U=cI₁ and c=±1. Here I₁ and I₂ denote the identity maps on H₁ and H₂, respectively.     

New Orlicz-Hardy spaces associated with divergence form elliptic operators

Let L be the divergence form elliptic operator with complex bounded measurable coefficients, ω the positive concave function on (0,∞) of strictly critical lower type pω∈(0,1] and ρ(t)=t−1/ω−1(t−1) for t∈(0,∞). In this paper, the authors study the Orlicz-Hardy space Hω,L(Rn) and its dual space BMOρ,L^*(Rn), where L^* denotes the adjoint operator of L in L²(Rn). Several characterizations of Hω,L(Rn), including the molecular characterization, the Lusin-area function characterization and the maximal function characterization, are established. The ρ-Carleson measure characterization and the John-Nirenberg inequality for the space BMOρ,L(Rn) are also given. As applications, the authors show that the Riesz transform ∇L−1/2 and the Littlewood-Paley g-function gL map Hω,L(Rn) continuously into L(ω). The authors further show that the Riesz transform ∇L−1/2 maps Hω,L(Rn) into the classical Orlicz-Hardy space Hω(Rn) for and the corresponding fractional integral L−γ for certain γ>0 maps Hω,L(Rn) continuously into , where is determined by ω and γ, and satisfies the same property as ω. All these results are new even when ω(t)=tp for all t∈(0,∞) and p∈(0,1).     

Viscosity approximation methods for nonexpansive nonself-mappings

Let E a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E^∗, and K be a closed convex subset of E which is also a sunny nonexpansive retract of E, and be nonexpansive mappings satisfying the weakly inward condition and F(T)≠∅, and be a fixed contractive mapping. The implicit iterative sequence {xt} is defined by for t∈(0,1)

xt=P(tf(xt)+(1−t)Txt).     

Convergence theorems for fixed points of uniformly continuous generalized Φ-hemi-contractive mappings

Let E be a real normed linear space, K be a nonempty subset of E and be a uniformly continuous generalized Φ-hemi-contractive mapping, i.e., , and there exist x^∗∈F(T) and a strictly increasing function , Φ(0)=0 such that for all x∈K, there exists j(x−x^∗)∈J(x−x^∗) such that

〈Tx−x^∗,j(x−x^∗)〉?‖x−x^∗‖2−Φ(‖x−x^∗‖).     

Weakly continuous holomorphic functions on Banach spaces with a shrinking and unconditional basis

Let E and F be complex Banach spaces, and let U be an open ball in E. We show that if E has a shrinking and unconditional basis, then every holomorphic function that is weakly continuous on U-bounded sets is weakly uniformly continuous on U-bounded sets.