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Let Ω ⊆ ℝn be a non-empty open bounded set and h: Ω → ℝ be a non-negative continuous function. We prove that for any u ∈ C2(Ω) ∩ C1($$\overline{\Omega}$$) solution of the Monge–Ampere equation $${\rm{det}}(D^2u)=h\;\;{\rm{in}}\;\Omega,$$, then ∇u satisfies the convex hull property: ∇u(Ω) ⊆ conv(∇u(∂Ω)).

For every Banach space $X$ with a Schauder basis consider the Banach algebra $\mathbb{R} I\oplus\mathcal{K}_\mathrm{diag}(X)$ of all diagonal operators that are of the form $\lambda I + K$. We prove that $\mathbb{R} I\oplus\mathcal{K}_\mathrm{diag}(X)$ is a Calkin algbra i.e., there exists a Banach space $\mathcal{Y}_X$ so that the Calkin algebra of $\mathcal{Y}_X$ is isomorphic as a Banach algebra to $\mathbb{R} I\oplus\mathcal{K}_\mathrm{diag}(X)$. Among other applications of this theorem we obtain that certain hereditarily indecomposable spaces and the James spaces $J_p$ and their duals endowed with natural multiplications are Calkin algebras, that all non-reflexive Banach spaces with unconditional bases are isomorphic as Banach spaces to Calkin algebras, and that sums of reflexive spaces with unconditional bases with certain James-Tsirelson type spaces are isomorphic as Banach spaces to Calkin algebras., Comment: 49 pages

We prove that if a Banach space admits a biorthogonal system whose dual part is norming, then the set of norm-attaining functionals is lineable. As a consequence, if a Banach space admits a biorthogonal system whose dual part is bounded and its weak-star closed absolutely convex hull is a generator system, then the Banach space can be equivalently renormed so that the set of norm-attaining functionals is lineable. Finally, we prove that every infinite dimensional separable Banach space whose dual unit ball is weak-star separable has a linearly independent, countable, weak-star dense subset in its dual unit ball. As a consequence, we show the existence of linearly independent norming sets which are not the dual part of a biorthogonal system.

In the present paper we study the problem of finding classical and weak solutions for a semilinear elliptic equation involving a singular term and a nonlinearity depending on the gradient. The approach we follow, based on a suitable combination of sub–supersolutions techniques and fixed point theory, is not quite the same in the two cases and requires different kind of assumptions.

It is shown that an infinite dimensional Banach space is hyper-barrelled if and only if it does not admit an infinite dimensional separable quotient.

The existence of Lagrange multipliers as a Radon measure is ensured for an elastic–plastic torsion problem associated to a nonlinear strictly monotone operator. A regularization of this result, namely the existence of L p Lagrange multipliers, is obtained under strong monotonicity assumption on the operator. Moreover, the relationships between elastic–plastic torsion problem and the obstacle problem are investigated. Finally, an example of the so-called “Von Mises functions” is provided, namely of solutions of the elastic–plastic torsion problem, associated to nonlinear monotone operators, which are not obtained by means of the obstacle problem in the case f = constant .

We prove that, for a suitable choice of real numbers $a, b$, every operator from $\ell_2$ to $X_{a,b}$ and from $X_{a,b}$ to $\ell_2$ must be compact, where $X_{a,b}$ is the Bourgain- Delbaen's space.

We gradually study i -operators on real vector spaces, on real topological vector spaces, and on real normed spaces. Among several things we prove the existence of real topological vector spaces (different from the James’ space) that are free of continuous i-operators. We also prove that every real normed space can be equivalently renormed to be free of norm i-operators. Examples of spaces of continuous functions not admitting norm i -operators and whose unit sphere is free of convex subsets with non-empty interior relative to it are also found. Finally, we also provide some results on a problem posed by Wenzel in [10]. Keywords : i -operator, Cartesian.

In the present paper we prove a multiplicity theorem for a quasi-linear elliptic problem with dependence on the gradient ensuring the existence of a positive solution and of a negative solution. In addition, we show the existence of the extremal constant-sign solutions: the smallest positive solution and the biggest negative solution. Our approach relies on extremal solutions for an auxiliary parametric problem. Other basic tools used in our paper are sub-supersolution techniques, Schaefer’s fixed point theorem, regularity results and strong maximum principle. In our hypotheses we only require a general growth condition with respect to the solution and its gradient, and an assumption near zero involving the first eigenvalue of the negative $$p$$ -Laplacian operator.

Throughout this paper, the authors introduce a new condition, defined by Assumption S ′ , which establishes a necessary and sufficient condition for the validity of the strong duality between a convex optimization problem and its Lagrange dual. This work will be focused on the context of emptiness of the interior of the ordering cone and convexity of the equality constraints. Moreover, this new condition will be necessary and sufficient for the infinite dimensional Lagrange multiplier rule. This new principle will find application to the elastic–plastic torsion problem, to the continuum model of transportation and to a problem with quadratic equality constraint with connected to evolutionary illumination and visibility problems.

In this paper we investigate the nature of family of pairs of separable Banach spaces (X, Y) such that $\mathcal{K}(X,Y)$ is complemented in $\mathcal{L}(X,Y)$. It is proved that the family of pairs (X,Y) of separable Banach spaces such that $\mathcal{K}(X,Y)$ is complemented in $\mathcal{L}(X,Y)$ is not Borel, endowed with the Effros–Borel structure.

Let A be a C ∗ -algebra. We prove that a holomorphic function of bounded type f : A → C is orthogonally additive on Asa if, and only if, it is additive on elements having zeroproduct if, and only if, there exist a positive functional φ in A∗ , a sequence (ψn) in L1(A∗∗,φ) and a power series holomorphic function h in Hb(A,A∗) such that h(a) = ∞ ∑ k=1 ψk ·ak and f (a) = 〈1A∗∗ ,h(a)〉 = ∫ h(a) dφ , for every a in A , where 1 A∗∗ denotes the unit element in A ∗∗ and L1(A∗∗,φ) is a noncommutative L1 -space. Mathematics subject classification (2010): Primary 46G20, 46L05; Secondary 46L51, 46E15, 46E50.

Elements a and b of a non-commutative Lp(M,τ) space associated to a von Neumann algebra, M, equipped with a normal semi-finite faithful trace τ, are called orthogonal if l(a)l(b)=r(a)r(b)=0, where l(x) and r(x) denote the left and right support projections of x. A linear map T from Lp(M,τ) to a normed space X is said to be orthogonality-to-p-orthogonality preserving if ∥T(a)+T(b)∥p=∥a∥p+∥b∥p whenever a and b are orthogonal. In this paper, we prove that an orthogonality-to-p-orthogonality preserving linear bijection from Lp(M,τ) to a Banach space is automatically continuous if 1≤p

We continue the study of the w-right and strong* topologies in general Banach spaces started in 36, 37 and 35. We show that in L1(μ)-spaces the w-right convergence of sequences admits a simpler control. Some considerations about these topologies will be contemplated in the particular cases of C*-algebras and JB*-triples in connection with summing operators. We also study (sequential) w-right-norm and strong*-norm continuity for holomorphic mappings. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

We prove that the Fremlin tensor product of two Banach lattices with the Radon-Nikodym property, with one of them atomic, has the Radon-Nikodym property.Keywords: Radon-Nikodym property, Fremlin tensor norm of Banach lattices.Quaestiones Mathematicae 30(2007), 45–56

We show that not all infinite dimensional closed subspaces of \({\ell_\infty}\) satisfy that the set of their norm-attaining functionals is lineable and prove that any closed subspace of \({\ell_\infty}\) containing c0 has the property that the set of their norm-attaining functionals is lineable.

For every well founded tree $\mathcal{T}$ having a unique root such that every non-maximal node of it has countable infinitely many immediate successors, we construct a $\mathcal{L}_\infty$-space $X_{\mathcal{T}}$. We prove that for each such tree $\mathcal{T}$, the Calkin algebra of $X_{\mathcal{T}}$ is homomorphic to $C(\mathcal{T})$, the algebra of continuous functions defined on $\mathcal{T}$, equipped with the usual topology. We use this fact to conclude that for every countable compact metric space $K$ there exists a $\mathcal{L}_\infty$-space whose Calkin algebra is isomorphic, as a Banach algebra, to $C(K)$., 27 pages, this version contains an improved result

In this article, the strong duality is treated. It is shown that the strong duality is equivalent to the non-emptiness of the subdifferential of a sort map involving the constraint functions. It is also noted that this technique is useful to verify the Assumption S. Indeed, the linearity of a constraint function h is not required as usually seen in the literature. Moreover, it is shown that this condition is easer to verify in the applications. We apply this new principle to the bi-obstacle problem, to the elastic-plastic torsion problem and to the continuum model of transportation.

We study holomorphic maps between C * -algebras A and B, when f: BA (0, ρ) → B is a holomorphic mapping whose Taylor series at zero is uniformly converging in some open unit ball U = BA (0, δ). If we assume that f is orthogonality preserving and orthogonally additive on A s a ∩ U and f (U) contains an invertible element in B, then there exist a sequence (hn) in B * * and Jordan * -homomorphisms Θ, Θ: M (A) → B * * such that f (x) = ∑ n = 1 ∞ h n Θ (an) = ∑n = 1 ∞ Θ (an) hn uniformly in a ∈ U. When B is abelian, the hypothesis of B being unital and f (U) ∩ i n v (B) ≠ ∅ can be relaxed to get the same statement., The authors are partially supported by the Spanish Ministry of Economy and Competitiveness, D.G.I. Project no. MTM2011-23843, and Junta de Andalucía Grant FQM3737.

It is proved that if X is innite-dimension al, then there exists an innite- dimensional space of X-valued measures which have innite variation on sets of positive Lebesgue measure. In term of spaceability, it is also shown that ca(B;;X ) n M , the measures with non- -nite variation, contains a closed subspace. Other considerations concern the space of vector measures whose range is neither closed nor convex. All of those results extend in some sense theorems of Mu~ noz Fern andez et al. (Linear Algebra Appl. 428 (2008)).

Let C be a countable subset of co that lies in the range of a (c 0)**-valued measure, then C lies in the range of a co-valued measure. We extend this result to C(K), where K is a compact Hausdorff space, i.e., we let C be a countable subset of C(K) that lies in the range of a C(K)**-valued measure, then C lies in the range of a C(K)-valued measure. We will also see that in any separable Banach space the result still holds

We show that if X is an L 1-space (resp. L ∞-space) and Y a Banach space, then X ⊗∧ Y (resp. X ⊗∨ Y) has the local unconditional structure (l.u.st.) property (or GL-property) if Y does.

We show, among other results, that if lambda denotes the Lebesgue measure on the Borel sets in [0, 1] and X is an infinite dimensional Banach space, then the set of measures whose range is neither closed nor convex is lineable in ca(lambda, X). We also show that, in certain situations, we have lineability of the set of X-valued and non-sigma-finite measures with relatively compact range. The lineability of sets of the type L-p(I)\L-q (I) is studied and some open questions are proposed. Some classical techniques together with the converse of the Lyapunov Convexity Theorem are used.

We show that, in certain situations, we have lineability in the set of bounded linear and non-absolutely summing operators. Examples on lineability of the set Pi(p)(E, F) \ I-p(E, F) are also presented and some open questions are proposed.

Let X and Y be two Banach spaces. In this short note we show that every weakly compact subset in the projective tensor product of X and Y can be written as the intersection of finite unions of sets of the form co ¯ ( K X ⊗ K Y ) , where K X and K Y are weakly compacts subsets of X and Y , respectively. If either X or Y has the Dunford–Pettis property, then any intersection of sets that are finite unions of sets of the form co ¯ ( K X ⊗ K Y ) , where K X and K Y are weakly compact sets in X and Y , respectively, is weakly compact.