Compact Hankel operators with conjugate analytic symbols

In 1986 S. Axler [3] proved that forf∈L _a ² the Hankel operator\(H_{\bar f} :L_a^2 \to (L^2 )^ \bot \) is compact if and only iff is in the little Bloch space {itB}{in0}. In this note we show that the same is true for\(H_{\bar f} :L_a^p \to L^p \), 1<p<∞. Moreover we prove that\(H_{\bar f} :L_a^1 \to L^1 \) is ?-compact if and only if\(|f'(z)|(1 - |z|^2 )\log \tfrac{1}{{1 - |z|^2 }} \to 0\) as |z|→1^?.     

Applicabilità proiettiva di due superficie

Let Ω ? ? ⁿ be a convex bounded open set, of class\(C^2 ,Q_\tau = \Omega \times \left[ {\tau ,\tau + T} \right],\tau \in \mathbb{R},T > 0.\). LetB be a linear continuous operator ofL ²Ω ? ? ^N inL ²Ω ? ? ^N . It is shown that if\(f \in L^2 (Q_\tau ,\mathbb{R}^N )\) then there exists a unique solution of the problem:\(u \in W^{2,1} (Q_\tau ,\mathbb{R}^N ),\alpha (x,t,H(u)) - \frac{{\partial u}}{{\partial t}} = f(x,t)\), in\(Q_\tau \), such thatu(x,t)=B u(x, τ+T) in Ω, wherea(x, t, ζ) is misurable in(x,t), continuous in ζ,a(x,t, 0)=0, and verifies condition (A). IfB=Id this is the classical periodic problem. If moreovera(x,t,ζ)=a(x,t+T, ζ) anda(x,t, H (Bu))=B a(x,t,H (u)) ?t ∈ ?, the analogous problem in Ω × ? is studied.     

On the spacel P(β)

Let\(\{ \beta (n)\} _{n = 0}^\infty \) be a sequence of positive numbers and 1 ≤p < ∞. We consider the spacel ^P(β) of all power series\(f(z) = \sum\limits_{n = 0}^\infty {\hat f(n)z^n } \) such that\(\sum\limits_{n = 0}^\infty {|\hat f(n)|^p |\beta (n)|^p } \). We give a necessary and sufficient condition for a polynomial to be cyclic inl ^P(β) and a point to be bounded point evaluation onl ^P(β).     

Stability of properly efficient points and isolated minimizers of constrained vector optimization problems

In this paper the constrained vector optimization problem mic _C f(x), g(x) ? ? K, is considered, where\(f:\mathbb{R}^n \to \mathbb{R}^m \) and\(g:\mathbb{R}^n \to \mathbb{R}^p \) are locally Lipschitz functions and\(C \subset \mathbb{R}^m \) and\(K \subset \mathbb{R}^p \) are closed convex cones. Several solution concepts are recalled, among them the concept of a properly efficient point (p-minimizer) and an isolated minimizer (i-minimizer). On the base of certain first-order optimalitty conditions it is shown that there is a close relation between the solutions of the constrained problem and some unconstrained problem. This consideration allows to “double” the solution concepts of the given constrained problem, calling sense II optimality concepts for the constrained problem the respective solutions of the related unconstrained problem, retaining the name of sense I concepts for the originally defined optimality solutions. The paper investigates the stability properties of thep-minimizers andi-minimizers. It is shown, that thep-minimizers are stable under perturbations of the cones, while thei-minimizers are stable under perturbations both of the cones and the functions in the data set. Further, it is shown, that sense I concepts are stable under perturbations of the objective data, while sense II concepts are stable under perturbations both of the objective and the constraints. Finally, the so called structural stability is discused.     

Multilicity results for some nonlinear ellitic roblems with asymtotically $${{\varvec{}}}$$-linear terms

Taking any \(p > 1\), we consider the asymptotically p-linear problem

$$\begin{aligned} \left\{ \begin{array}{ll} - {{\mathrm{div}}}(a(x,u,\nabla u)) + A_t(x,u,\nabla u)\ = \ \lambda ^\infty |u|^{p-2}u + g^\infty (x,u) &{}\quad \hbox {in}\;\Omega ,\\ u\ = \ 0 &{}\quad \hbox {on}\;\partial \Omega , \end{array} \right. \end{aligned}$$

where \(\Omega \) is a bounded domain in \(\mathbb R^N\), \(N\ge 2\), \(A(x,t,\xi )\) is a real function on \(\Omega \times \mathbb R\times \mathbb R^N\) which grows with power p with respect to \(\xi \) and has partial derivatives \(A_t(x,t,\xi ) = \frac{\partial A}{\partial t}(x,t,\xi )\), \(a(x,t,\xi ) = \nabla _\xi A(x,t,\xi )\). If \(A(x,t,\xi ) \rightarrow A^\infty (x,t)\) and \(\frac{g^\infty (x,t)}{|t|^{p-1}} \rightarrow 0\) as \(|t| \rightarrow +\infty \), suitable assumptions, variational methods and either the cohomological index theory or its related pseudo-index one, allow us to prove the existence of multiple nontrivial bounded solutions in the non-resonant case, i.e. if \(\lambda ^\infty \) is not an eigenvalue of the operator associated to \(\nabla _\xi A^\infty (x,\xi )\). In particular, while in [14] the model problem \(A(x,t,\xi ) = \mathcal{A}(x,t) |\xi |^p\) with \(p > N\) is studied, here our goal is twofold: extending such results not only to a more general family of functions \(A(x,t,\xi )\), but also to the more difficult case \(1 < p \le N\).

     

Gruppi che ammettono un automorfismo 4-spezzante

LetG be a group admitting a 4-splitting automorphism (i.e. an automorphism σ such that\(gg^\sigma g^{\sigma ^2 } g^{\sigma ^3 } = 1\) for everyg∈G). In this paper we prove that ifG≠1 is solvable with derived lengthd thenG′ is nilpotent of class not greater than (4 ^d?1?1)/3.     

Some subclasses of the real-valued honorary Baire two functions on ℝ n

Certain subclasses of the class of Baire one real-valued functions have very nice properties, especially concerning their points of continuity and their preservation of connectedness for many connected sets. A Gibson [weakly Gibson] is defined by the requirement that \(f(\overline{U})\subseteq\overline{f(U)}\) for every open [open connected] set U?? ⁿ . It is known that Baire one, Gibson functions are continuous, and that Baire one, weakly Gibson functions have Darboux-like properties in the sense that if U is an open connected set and \(U\subseteq S\subseteq\overline{U}\), then f(S) is an interval. Here we study the situation where the Baire one condition is replaced by honorary Baire two. Distinctly different results are found.     

Principal parts bundles on projective spaces and quiver representations

We study the principal parts bundles \(\mathcal {P}^{k}\mathcal {O}_{\mathbb {P}^{n}}(d)\) as homogeneous bundles and we describe their associated quiver representations. With this technique we show that if n≥2 and 0≤d<k then there exists an invariant decomposition \(\mathcal {P}^{k}\mathcal {O}_{\mathbb {P}^{n}}(d)=Q_{k,d}\oplus(S^{d}V\otimes \mathcal {O}_{\mathbb {P}^{n}})\) with Q _k,d a stable homogeneous vector bundle. The decomposition properties of such bundles were previously known only for n=1 or k≤d or d<0. Moreover we show that the Taylor truncation maps \(H^{0}\mathcal {P}^{k}\mathcal {O}_{\mathbb {P}^{n}}(d)\to H^{0}\mathcal {P}^{h}\mathcal {O}_{\mathbb {P}^{n}}(d)\), defined for any h≤k and any d, have maximal rank.     

Witt equivalence of central simple algebras with involution

The notion of Witt equivalence of central simple algebras with involution is introduced. It is shown that the standard invariants, i.e. the discriminant, the signature and the Clifford algebra, depend only on the Witt class of the algebra with involution. For a given filedF the tensor product is used to construct a semigroup\(\tilde S\left( F \right)\) and this semigroup is shown to have properties analogous to the multiplicative properties of the Witt ring of quadratic forms overF.     

Upper triangular matrix operators with diagonal (T_1,T_2), T_2 k-nilpotent

If \(T=\left(\begin{array}{clcr}T_1&\quad C\\ 0&\quad T_2\end{array}\right) \in B(\mathcal{X }_1\oplus \mathcal{X }_2)\) is a Banach space upper triangular operator matrix with diagonal \((T_1, T_2)\) such that \(T_2\) is \(k\)-nilpotent for some integer \(k\ge 1\), then \(T\) inherits a number of its spectral properties, such as SVEP, Bishop’s property \((\beta )\) and the equality of Browder and Weyl spectrum, from those of \(T_1\). This paper studies such operators. The conclusions are then applied to provide a general framework for results pertaining (for example) to Browder, Weyl type theorems and supercyclicity for classes of Hilbert space operators, such as \(k\)-quasi hyponormal, \(k\)-quasi isometric and \(k\)-quasi paranormal operators, defined by a positivity condition.     

The representation type of rational normal scrolls

The goal of this paper is to demonstrate that all non-singular rational normal scrolls \(S(a_0,\ldots ,a_k)\subseteq \mathbb P ^N\), \(N =\sum _{i=0}^k(a_i)+k\), (unless \(\mathbb P ^{k+1}=S(0,\ldots ,0,1)\), the rational normal curve \(S(a)\) in \(\mathbb P ^a\), the quadric surface \(S(1,1)\) in \(\mathbb P ^3\) and the cubic scroll \(S(1,2)\) in \(\mathbb P ^4\)) support families of arbitrarily large rank and dimension of simple Ulrich (and hence indecomposable ACM) vector bundles. Therefore, they are all of wild representation type unless \(\mathbb P ^{k+1}\), \(S(a)\), \(S(1,1)\) and \(S(1,2)\) which are of finite representation type.     

Lie ideals and (\alpha ,\beta )-derivations of *-prime rings

Let \((R,*)\) be a \(2\)-torsion free \(*\)-prime ring with involution \(*\), \(L\ne \{0\}\) be a \(*\)-Lie ideal of \(R\). An additive mapping \(d:R\rightarrow R\) is called an \((\alpha ,\beta )\)-derivation of \(R\) such that \(d(xy)=d(x)\alpha (y)+\beta (x)d(y)\). In the present paper, we shall show that when \(L\) satisfies any of several identities involving \(d\), then \(L\) is central.     

On the graded central polynomials for elementary gradings in matrix algebras

Let \(F\) be a field of characteristic zero. Let \(M_{n}(F)\) be the algebra of all \(n \times n\) matrices over \(F\). We have found, in this article, a generating set for the graded central polynomials of \(M_{n}(F)\) when it is equipped with an elementary grading whose neutral component coincides with the diagonal.     

Factorization in Hurwitz series domain

Let A be a commutative ring with unit and HA the set of formal expressions of the type \(f=\sum_{i:0}^{\infty}a_{i}X^{i}\) where a _i ∈A. When \(g=\sum_{i:0}^{\infty}b_{i}X^{i}\) then \(f+g=\sum_{i:0}^{\infty}(a_{i}+b_{i})X^{i}\) and \(f*g=\sum_{n:0}^{\infty}c_{n}X^{n}\) with \(c_{n}=\sum_{i:0}^{n}C_{n}^{i}a_{i}b_{n-i}\), where \(C_{n}^{i}={n!\over i!(n-i)!}\). With these two operations HA is a commutative ring with identity. It was introduced and studied by Keigher in 1997. In this note we continue the investigation and we focus on factorization in HA and its sub-ring hA of Hurwitz polynomials. We recall from Benhissi (Contrib. Algebra. Geom. 48(1):251–256, 2007, Proposition 1.1) and Keigher (Commun. Algebra 25(6):1845–1859, 1997, Corollary 2.8) that HA is an integral domain if and only if A is an integral domain with zero characteristic. Let π ₀:HA?A be the natural ring homomorphism that assigns to each series its constant term. The key property is that a series f∈HA is a unit in HA if and only if π ₀(f) is a unit in A, Keigher (Commun. Algebra 25(6):1845–1859, 1997, Proposition 2.5).     

Complexification in \mathcal{A}-modules

As for the classical complexification of real vector spaces, the sheaf-theoretic version shows that free \(\mathcal{A}\)-modules of finite rank, with \(\mathcal{A}\) an ordered nonzero-nilsquare free ?-algebra sheaf, admits a complex structure if and only if there exists on \(\mathcal{E}\) an \(\mathcal{A}\)-automorphism J such that J ²=?I.     

Martin’s Axiom and separated mad families

Two families \(\mathcal{A}, \mathcal{B}\) of subsets of ω are said to be separated if there is a subset of ω which mod finite contains every member of \(\mathcal{A}\) and is almost disjoint from every member of \(\mathcal{B}\). If \(\mathcal{A}\) and \(\mathcal{B}\) are countable disjoint subsets of an almost disjoint family, then they are separated. Luzin gaps are well-known examples of ω ₁-sized subfamilies of an almost disjoint family which can not be separated. An almost disjoint family will be said to be ω ₁-separated if any disjoint pair of ≤ω ₁-sized subsets are separated. It is known that the proper forcing axiom (PFA) implies that no maximal almost disjoint family is ≤ω ₁-separated. We prove that this does not follow from Martin’s Axiom.     

Some remarks on spectral states ofLMC-algebras

We generalize the notion of a spectral state (as introduced for Banach algebras by Moore, Bonsall and Duncan) to the context of locally multiplicatively-convex (LMC) algebras by proceeding in a way analogous to the generalization of numerical range theory from Banach algebras toLMC-algebras carried out by Giles and Koehler. Among the results obtained in this note are integral representations of spectral states by probability measures on the structure space ofA and the determination of the extreme points of the convex set\(\Omega _A \) of all spectral states on a commutativeLMC-algebraA (which is related to different Choquet boundaries) as well as a characterization of symmetric involutions by the coincidence of the notions of positive state and spectral state and a characterization of theQ-property by the weak-^*-boundedness of\(\Omega _A \). The paper finishes with two elementary commutativity criteria involving spectral states and two Korovkin-type theorems for the approximation of unital algebra homomorphisms by σ-equicontractive nets of linear operators mapping anLMC-algebraA into theLMC-algebra of all continuous complex-valued functions on a completely regular spaceX.     

Strong extensions for q-summing o perators acting in p-convex Banach function s paces for $$1 \le p \le q$$1≤ p≤ q

Let \(1\le p\le q<\infty \) and let X be a p-convex Banach function space over a \(\sigma \)-finite measure \(\mu \). We combine the structure of the spaces \(L^p(\mu )\) and \(L^q(\xi )\) for constructing the new space \(S_{X_p}^{\,q}(\xi )\), where \(\xi \) is a probability Radon measure on a certain compact set associated to X. We show some of its properties, and the relevant fact that every q-summing operator T defined on X can be continuously (strongly) extended to \(S_{X_p}^{\,q}(\xi )\). Our arguments lead to a mixture of the Pietsch and Maurey-Rosenthal factorization theorems, which provided the known (strong) factorizations for q-summing operators through \(L^q\)-spaces when \(1 \le q \le p\). Thus, our result completes the picture, showing what happens in the complementary case \(1\le p\le q\).     

Periodic orbits for a class ofC 1 three-dimensional systems

We studyC ¹ perturbations of a reversible polynomial differential system of degree 4 in\(\mathbb{R}^3 \). We introduce the concept of strongly reversible vector field. If the perturbation is strongly reversible, the dynamics of the perturbed system does not change. For non-strongly reversible perturbations we prove the existence of an arbitrary number of symmetric periodic orbits. Additionally, we provide a polynomial vector field of degree 4 in\(\mathbb{R}^3 \) with infinitely many limit cycles in a bounded domain if a generic assumption is satisfied.     

Lower bound for matrix operators on the Euler weighted sequence space e_{w,p}^{\theta} (0<p<1)

Let A=(a _n,k ) _n,k≥0 be a non-negative matrix. Denote by \(L_{l_{p} (w),~e_{w,q}^{\theta}}(A)\) the supremum of those L, satisfying the following inequality: