Tensor products,asymptotic intertwinings and property (R)

A Banach space operator satisfies property (R) if the set of its finite multiplicity isolated eigenvalues equals the set of its finite rank left poles. Property (R) transfers from operators A and B to their tensor product A?B. Asymptotic intertwining by the identity operator does not preserve property (R). Additional hypotheses are required.     

On convergence of sections of sequences in Banach spaces

An elementary proof of the (known) fact that each element of the Banach space? _w ^p (X) of weakly absolutelyp-summable sequences (if 1≤p<∞) in the Banach spaceX is the norm limit of its sections if and only if each element of? _w ^p (X) is a norm null sequence inX, is given. Little modification to this proof leads to a similar result for a family of Orlicz sequence spaces. Some applications to spaces of compact operators on Banach sequence spaces are considered.     

Operateurs essentiellement reguliers dans les espaces de Banach

Given a closed operatorA acting in a Banach spaceX, we define the regular (respectively the essentialy regular) spectrum σ _r (A) (respectively σ _e,r (A)) ofA. We prove that σ _r (A) and σ _e,r (A) are a closed subsets of the classical spectrum σ(A) ofA. Morever ifA is bounded we prove that σ _r (A) and σ _e,r (A)) satisfies the spectral mapping theorem.     

On a conjecture concerning some automatic continuity theorems

Let A and B be commutative locally convex algebras with unit. A is assumed to be a uniform topological algebra. Let Φ be an injective homomorphism from A to B. Under additional assumptions, we characterize the continuity of the homomorphism Φ^?1/Im?Φ by the fact that the radical (or strong radical) of the closure of Im?Φ has only zero as a common point with Im?Φ. This gives an answer to a conjecture concerning some automatic continuity theorems on uniform topological algebras.     

On bowtie rings,universal survival rings and universal lying-over rings

If (R,M) is a quasilocal integral domain of Krull dimension n,?1<n≤∞, and E is the direct sum of denumerably many copies of R/M, then T:=R?E is a reduced n-dimensional universal survival ring which is not a universal lying-over ring. In fact, T is a new kind of such ring, as T is not (isomorphic to) an A+B construction and T is not a ring of continuous real-valued functions. The analysis includes identifying all the prime ideals of T and showing that T is its own total quotient ring and satisfies Property A. The assertion would fail if n=1, as T would be a universal lying-over ring in this case. It is also shown that a (commutative unital) ring A satisfies Property A if and only if each ideal of A that consists only of zero-divisors survives in the complete ring of quotients of A.     

The interior operation in aψ-density topology

We prove that for an arbitrary setA ? ? its interior in aψ-density topology equalsA ∩ φ^β(B), whereB is a measurable kernel ofA andβ is some countable ordinal. Moreover, eachβ, 1≤β<Ω, realizes the interior ofA for someA εS.     

Everywhere Hölder continuity of the spatial gradient of weak solutions to nonlinear parabolic systems

We consider weak solutions to the parabolic system ?u ⁱ?t?D _α A _i ^α (?u)=B _i(?u) in (i=1,...,) (Q=Ω×(0,T), R ⁿ a domain), where the functionsB _i may have a quadratic growth. Under the assumptionsn≤2 and ?u ?L _loc ^4+δ (Q; R ^nN ) (δ>0) we prove that ?u is locally Hölder continuous inQ.     

The role of convexity in existence theorems for invariant and hyperinvariant subspaces in hilbert space

We use extreme point methods to obtain some existence theorems for invariant and hyperinvariant subspaces for certain classes of operators In Hilbert space. These convexity methods have, heretofore, not been used in this context, however it would appear that they provide useful alternatives to existing methods. Using the Krein-Milman theorem we prove several variants of the classical lemma of I. Schur. In one of the main results we prove the following: LetA ∈B (H), thenA andA* have a common non-trivial closed hyperinvariant subspace if and only if there exists a non-scalar operatorW εB(H) such that bothW andW* belong to the double commutant ofA.     

Calcul Fonctionnel Holomorphe Global Unicité et Invariance Par Les Homomorphismes D’algebres de Banach

LetA be a commutative Banach algebra with unit. Denote byX _A, the global spectrum ofA. There is a holomorphic functional calculusθ _A:O(X _A)→A such thatθ _A(â)=a. In this paper, we show the uniqueness of the global holomorphic functional calculus and we establish its compatibility with Banach algebra morphisms. We also extend this holomorphic functional calculus to the case ofImc algebras.     

Sur la Topologiem-Convexe d’une algèbre localementA-convexe

In a locallyA-convex algebra (E, τ) we consider the associatedm-convex topologym(τ). We show that the completion ofE with respect tom(τ) is always a locallyA-convex algebra contained in the complete locally convex space obtained from (E, τ). The topologym(τ) is also used to characterize locally boundedly multiplicatively convex algebras among locallyA-convex ones.     

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).     

Distance and fractional isomorphism in Steiner triple systems

In [8], Quattrochi and Rinaldi introduced the idea ofn ^?1-isomorphism between Steiner systems. In this paper we study this concept in the context of Steiner triple systems. The main result is that for any positive integerN, there existsv ₀(N) such that for all admissiblev≥v ₀(N) and for each STS(v) (sayS), there exists an STS(v) (sayS′) such that for somen>N, S is strictlyn ^?1-isomorphic toS′. We also prove that for all admissiblev≥13, there exist two STS(v)s which are strictly 2^?1-isomorphic. Define the distance between two Steiner triple systemsS andS′ of the same order to be the minimum volume of a tradeT which transformsS into a system isomorphic toS′. We determine the distance between any two Steiner triple systems of order 15 and, further, give a complete classification of strictly 2^?1-isomorphic and 3^?1-isomorphic pairs of STS(15)s.     

Paranormal and (p,k)-quasihyponormal operators: Spectral continuity

Continuity of the set theoretic functions spectrum, Weyl spectrum, Browder spectrum and essential surjectivity spectrum on the classes consisting of (p, k)-quasihyponormal,M-hyponormal, totally *-paranormal and paranormal (Hilbert space) operators is proved.     

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.     

Some extensions of the Laplace problem

Let ?₁(a,b) be a regular lattice with the fundamental cell a rectangle with sides a,b, let ?₂(a,b,?) be a regular lattice with the fundamental cell a parallelogram with sides a,b and angle ? and let ?₃(a,b,c) be a regular lattice with the fundamental cell \(C_{0}^{( 3) }\) as in Fig. 7. In this paper we compute the probability that a random rectangle r of constant side l,m intersects a side of the lattice. In particular when the rectangle r becomes a segment of length l, (m=0) we obtain the Laplace probability.     

On a class of hyperbolic 3-orbifolds of small volume and small heegaard genus associated to 2-bridge links

We consider a class of hyperbolic 3-orbifoldsO(α/β); the underlying topological space of such an orbifold is the 3-sphere and the singular set is obtained by adding the two standard (upper and lower) unknotting tunnels to a 2-bridge linkL(α/β) (and associating branching order two to both unknotting tunnels). These 3-orbifolds are extremal with respect to the notion of Heegaard genus or Heegaard number of 3-orbifolds; it is to be expected that they are also extremal with respect to the volume, that is the smallest volume hyperbolic 3-orbifolds should belong to this or some closely related class. We show that an orbifoldO(α/β) has a uniqueD ₂-covering by an orbifold? _n(α/β) wose space is the 3-sphere and whose singular set is the same 2-bridge linkL(α/β) used for the construction ofO(α/β); moreoverO(α/β) is hyperbolic if and only if? _n(α/β) is hyperbolic. As the volumes of the orbifolds? _n(α/β) are known resp. can be computed, this allows to compute the volumes of the orbifoldsO(α/β). The problem of computation of volumes remains open for some closely related classes of 3-orbifolds which are also extremal with respect to the Heegaard genus (for example associating a branching order bigger than two to one or both unknotting tunnels).     

Refining Castelnuovo-Halphen bounds

Fix integers r,d,s,π with r≥4, d?s, r?1≤s≤2r?4, and π≥0. Refining classical results for the genus of a projective curve, we exhibit a sharp upper bound for the arithmetic genus p _a (C) of an integral projective curve C?? ^r of degree d, assuming that C is not contained in any surface of degree <s, and not contained in any surface of degree s with sectional genus >π. Next we discuss other types of bound for p _a (C), involving conditions on the entire Hilbert polynomial of the integral surfaces on which C may lie.     

Feebly associativeP-hypergroupoids

In this paper we study the hyperstructures, saidP-hypergroupoids, (H, P*) in whichH is a set andP* is one of the hyperoperations defined as follows: ?(x, y) εH ²,xP*y=xyP orxP*y=Pxy whereP is a subset ofH. In particular we give a general formula for to express the simple hyperproducts ofn elements and then we consider some cases in which (H, P*) is feebly associative. We study, in such cases theβ-relations.     

Sui gruppi unitari finitari e sulla congettura di dieudonné

In this paper we consider some questions concerning unitary spaces (V, h), even though (V, h) is not finitely generated. Our main result is as follows: letF be an infinite field of characteristic ≠2 andD anm ²-dimensional central division algebra overF with an involutionj≠1. Let Σ_j(D) denote the subgroup of the multiplicative groupD ^* generated by the non zero symmetric elements. If (V, h) is an infinite dimensional regular unitary space of Witt index at least two overD, then the finitary unitary groupFU(h) is a simple group if and only ifD*=Σ_j(D)[D*, D*]. On the other hand, when (V, h) is not regular,FU(h) cannot be simple since it containsFU ₀(h), the subgroup of elements ofFU(h) acting trivially onrad(V, h), as a normal subgroup. In the non regular case we show that under the above assumptions evenFU ₀(h) is not a simple group.     

Wijsman topology on function spaces

LetC(X,Y) be the space of continuous functions from a metric space (X,d) to a metric space (Y, e).C(X, Y) can be thought as subset of the hyperspaceCL(X×Y) of closed and nonempty subsets ofX×Y by identifying each element ofC(X,Y) with its graph. We considerC(X,Y) with the topology inherited from the Wijsman topology induced onCL(X×Y) by the box metric ofd ande. We study the relationships between the Wijsman topology and the compact-open topology onC(X,Y) and also conditions under which the Wijsman topology coincide with the Fell topology. Sufficient conditions under which the compactopen topology onC(X,Y) is weaker than the Wijsman topology are given (IfY is totally bounded, then for every metric spaceX the compactopen topology onC(X,Y) is weaker than the Wijsman topology and the same is true forX locally connected andY rim-totally bounded). We prove that a metric spaceX is boundedly compact iff the Wijsman topology onC(X, ?) is weaker than the compact-open topology. We show that ifX is a σ-compact complete metric space andY a compact metric space, then the Wijsman topology onC(X,Y) is Polish.