Let Ω be a measure space, and E, F be separable Banach spaces. Given a multifunction $f:\Omega \times E\to 2^F$, denote by $N_f\left(x\right)$ the set of all measurable selections of the multifunction $f(·,x\left(·\right)):\Omega \to 2^F$, s ↦ f(s,x(s)), for a function x: Ω → E. First, we obtain new theorems on H-upper/H-lower/lower semicontinuity (without assuming any conditions on the growth of the generating multifunction f(s,u) with respect to u) for the multivalued (Nemytskiĭ) superposition operator $N_f$ mapping some open domain G ⊂ X into $2^Y$, where X and Y are Köthe-Bochner...

Viene studiata la semicontinuità rispetto alla topologia di $L^{\mathrm{\infty }}(\mathrm{\Omega };{𝐑}^m)$ per alcuni funzionali del Calcolo delle Variazioni dipendenti da funzioni a valori vettoriali.

We show that if $P_k$ is a boundedly complete, unconditional Schauder decomposition of a Banach space X, then X is weakly sequentially complete whenever $P_{k}X$ is weakly sequentially complete for each k ∈ ℕ. Then through semi-embeddings, we give a new proof of Lewis’s result: if one of Banach spaces X and Y has an unconditional basis, then X ⊗̂ Y, the projective tensor product of X and Y, is weakly sequentially complete whenever both X and Y are weakly sequentially complete.

We associate a $C^{*}$-algebra to a locally compact Hausdorff groupoid with the property that the range map is locally injective. The construction generalizes J. Renault’s reduced groupoid $C^{*}$-algebra of an étale groupoid and has the advantage that it works for the groupoid arising from a locally injective dynamical system by the method introduced in increasing generality by Renault, Deaconu and Anantharaman-Delaroche. We study the $C^{*}$-algebras of such groupoids and give necessary and sufficient conditions...

Soit $X$ un espace riemannien symétrique et $C_0\left(X\right)$ l’espace des fonctions continues sur $X$ tendant vers 0 à l’infini. On démontre qu’un opérateur $(D_{A^{\text{'}}},A)$, invariant par les isométries de $X$, engendre un semi-groupe fortement continu de contractions sur $C_0\left(X\right)$ s’il est dissipatif et si son domaine contient les fonctions de classe ${𝒞}^{\infty }$ à support compact.

The aim of the paper is two-fold. First, we investigate the ψ-Bessel potential spaces on ${ℝ}_{0+}^{n+1}$ and study some of their properties. Secondly, we consider the fractional powers of an operator of the form $-A_{\pm }=-\psi \left(D_{x^{\text{'}}}\right)\pm \partial /\left(\partial x_{n+1}\right)$, $(x^{\text{'}},x_{n+1})\in {ℝ}_{0+}^{n+1}$, where $\psi \left(D_{x^{\text{'}}}\right)$ is an operator with real continuous negative definite symbol ψ: ℝⁿ → ℝ. We define the domain of the operator $-{(-A_{\pm })}^{\alpha }$ and prove that with this domain it generates an $L_p$-sub-Markovian semigroup.