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

     

On the spectral radius of positive operators on Banach sequence spaces

Let K₁,…,K_n be (infinite) non-negative matrices that define operators on a Banach sequence space. Given a function f:[0,∞)×…×[0,∞)→[0,∞) of n variables, we define a non-negative matrix and consider the inequality

     

Noncommutative transforms and free pluriharmonic functions

In this paper, we study free pluriharmonic functions on noncommutative balls γ[Bn(H)], γ>0, and their boundary behavior. These functions have the form

     

Asymptotic behavior of coupled dynamical systems with multiscale aspects

We study the asymptotic behavior, as time variable t goes to +∞, of nonautonomous dynamical systems involving multiscale features. As a benchmark case, given H a general Hilbert space, and two closed convex functions, and β a function of t which tends to +∞ as t goes to +∞, we consider the differential inclusion

     

A generalized inductive limit strict topology on the space of bounded continuous functions

A generalized inductive limit strict topology β_∞ is defined on C_b(X, E), the space of all bounded, continuous functions from a zero-dimensional Hausdorff space X into a locally -convex space E, where is a field with a nontrivial and nonarchimedean valuation, for which is a complete ultrametric space. Many properties of the topology β_∞ are proved and the dual of (C_b (X, E), β_∞) is studied.     

Spectral theory for perturbed Krein Laplacians in nonsmooth domains

We study spectral properties for HK,Ω, the Krein-von Neumann extension of the perturbed Laplacian −Δ+V defined on , where V is measurable, bounded and nonnegative, in a bounded open set Ω⊂Rn belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class C1,r, r>1/2. In particular, in the aforementioned context we establish the Weyl asymptotic formula

     

A Robertson-type uncertainty principle and quantum Fisher information

Let A₁,…,A_N be complex self-adjoint matrices and let ρ be a density matrix. The Robertson uncertainty principle

     

Heights of algebraic numbers modulo multiplicative group actions

Given a number field K and a subgroup G⊂K^∗ of the multiplicative group of K, Silverman defined the G-height H(θ;G) of an algebraic number θ as

     

A sharp form of an embedding into exponential and double exponential spaces

Let Ω be a bounded domain in . In the well-known paper (Indiana Univ. Math. J. 20 (1971) 1077) Moser found the smallest value of K such that

     

Idéaux fermés d'algèbres de Beurling analytiques sur le bidisque

We study the closed ideal in the Beurling algebras of holomorphic function f in the bidisc such that

     

On the vertex index of convex bodies

We introduce the vertex index, vein(K), of a given centrally symmetric convex body K⊂Rd, which, in a sense, measures how well K can be inscribed into a convex polytope with small number of vertices. This index is closely connected to the illumination parameter of a body, introduced earlier by the first named author, and, thus, related to the famous conjecture in Convex Geometry about covering of a d-dimensional body by d² smaller positively homothetic copies. We provide asymptotically sharp estimates (up to a logarithmic term) of this index in the general case. More precisely, we show that for every centrally symmetric convex body K⊂Rd one has

     

Adams' inequalities for bi-Laplacian and extremal functions in dimension four

Let Ω⊂R⁴ be a smooth oriented bounded domain, be the Sobolev space, and be the first eigenvalue of the bi-Laplacian operator Δ². Then for any α: 0?α<λ(Ω), we have

     

The Bourgain ?-index of mixed Tsirelson space

Suppose that is a sequence of regular families of finite subsets of such that contains all singletons, and (θ_n)_n=1^∞ is a nonincreasing null sequence in (0,1). The mixed Tsirelson space is the completion of c₀₀ with respect to the implicitly defined norm

     

Note on separate continuity and the Namioka property

A pair 〈B,K〉 is a Namioka pair if K is compact and for any separately continuous , there is a dense A⊆B such that f is ( jointly) continuous on A×K. We give an example of a Choquet space B and separately continuous such that the restriction fΔ_| to the diagonal does not have a dense set of continuity points. However, for K a compact fragmentable space we have: For any separately continuous and for any Baire subspace F of T×K, the set of points of continuity of is dense in F. We say that 〈B,K〉 is a weak-Namioka pair if K is compact and for any separately continuous and a closed subset F projecting irreducibly onto B, the set of points of continuity of fF_| is dense in F. We show that T is a Baire space if the pair 〈T,K〉 is a weak-Namioka pair for every compact K. Under (CH) there is an example of a space B such that 〈B,K〉 is a Namioka pair for every compact K but there is a countably compact C and a separately continuous which has no dense set of continuity points; in fact, f does not even have the Baire property.     

Exposed sets in potential theory

Let U be a relatively compact open subset of a harmonic space, and H(U) be the function space of all continuous functions on which are harmonic on U. We give a complete characterization of the H(U)-exposed subsets of . This extends the results of [J. Lukeš, T. Mocek, M. Smr?ka, J. Spurný, Choquet like sets in function spaces, Bull. Sci. Math. 127 (2003) 397-437].     

Sobolev embeddings for Riesz potential spaces of variable exponents near 1 and Sobolev's exponent

Our aim in this paper is to deal with Sobolev embeddings for Riesz potentials of order α for functions f satisfying the Orlicz type condition

     

Dimension free estimates for discrete Riesz transforms on products of abelian groups

Let G be a lca group with a fixed g₀∈G, spanning an infinite subgroup. Let τ_j, acting on L²(Gⁿ), be translation by g_o in the jth coordinate; the discrete derivatives ∂_j=I−τ_j define a discrete Laplacian and discrete Riesz transforms . We get dimension-free estimates

     

Critical points between varieties generated by subspace lattices of vector spaces

We denote by the semilattice of all compact congruences of an algebra A. Given a variety V of algebras, we denote by the class of all semilattices isomorphic to for some A∈V. Given varieties V and W of algebras, the critical point of V under W is defined as . Given a finitely generated variety V of modular lattices, we obtain an integer ?, depending on V, such that for any n≥? and any field F.In a second part, using tools introduced in Gillibert (2009) [5], we prove that: