Some inequalities concerning the James constant in Banach spaces

Let A₂(X) be the constant introduced by Baronti, Casini and Papini. This paper discusses the constant A₂(X) and states an estimate in terms of the James constant. The estimate enables us to improve an inequality between the James and von Neumann-Jordan constants.     

An Extension of a Simply Inequality Between von Neumann–Jordan and James Constants in Banach Spaces

For a non-trivial Banach space X, let J(X), CNJ(X), C_(NJ)~(p)(X) respectively stand for the James constant, the von Neumann–Jordan constant and the generalized von Neumann–Jordan constant recently inroduced by Cui et al. In this paper, we discuss the relation between the James and the generalized von Neumann–Jordan constants, and establish an inequality between them: C_(NJ)~(p)(X) ≤J(X) with p ≥ 2, which covers the well-known inequality CNJ(X) ≤ J(X). We also introduce a new constant, from which we establish another inequality that extends a result of Alonso et al.     

On a new geometric constant related to the von Neumann-Jordan constant

The von Neumann-Jordan constant CNJ(X) is computed for X being ?₂−?₁ and ?_∞−?₁ space by introducing a new geometric constant γX(t). These partly give an answer to an open question posed by Kato et al. Some basic properties of this new coefficient are investigated. Moreover, we obtain a new class of Banach spaces with uniform normal structure.     

A simple inequality for the von Neumann-Jordan and James constants of a Banach space

Let CNJ(X) and J(X) be the von Neumann-Jordan and James constants of a Banach space X, respectively. We shall show that CNJ(X)?J(X), where equality holds if and only if X is not uniformly non-square. This answers affirmatively to the question in a recent paper by Alonso et al. [J. Alonso, P. Martín, P.L. Papini, Wheeling around von Neumann-Jordan constant in Banach spaces, Studia Math. 188 (2008) 135-150]. This inequality looks quite simple and covers all the preceding results. In particular this is much stronger than Maligranda's conjecture: .     

Perturbing the boundary conditions of the generator of a cosine family

Let A be a densely defined, closed linear operator (which we shall call maximal operator) with domain D(A) on a Banach space X and consider closed linear operators L:D(A)???X and ??:D(A)???X (where ?X is another Banach space called boundary space). Putting conditions on L and ??, we show that the second order abstract Cauchy problem for the operator A _?? with A _?? u=Au and domain D(A _??):={u??D(A):Lu=??u} is well-posed and thus it generates a cosine operator function on the Banach space X.     

On a generalized James constant

We introduce a generalized James constant J(a,X) for a Banach space X, and prove that, if J(a,X)<(3+a)/2 for some a∈[0,1], then X has uniform normal structure. The class of spaces X with J(1,X)<2 is proved to contain all u-spaces and their generalizations. For the James constant J(X) itself, we show that X has uniform normal structure provided that , improving the previous known upper bound at 3/2. Finally, we establish the stability of uniform normal structure of Banach spaces.     

A note on Jordan-von Neumann constant and James constant

Let X be a non-trivial Banach space. L. Maligranda conjectured CNJ(X)?1+J²(X)/4 for James constant J(X) and von Neumann-Jordan constant CNJ(X) of X. Satit Saejung gave a proof of it in 2006. In this note, we show that the last step in Satit Saejung's proof is not valid. Using his proof, the result should be . On the other hand, we give a new proof of CNJ(X)?1+J²(X)/4. As an application, we give a relation between J(X) and J(lp(X)).     

Existence and asymptotic behavior of a functional differential equation in Banach space

Existence and asymptotic behavior of solutions are given for the equation u′(t) = ?A(t)u(t) + F(t,u_t) (t ? 0) and u₀ = ? ? C([?r,0]; X)  C. The space X is a Banach space; the family $\text{{A(t) ¦ 0 ? t ? T}}$ of unbounded linear operators defined on D(A) ? X → X generates a linear evolution system and F: C → X is continuous with respect to a fractional power of A(t₀) for some t₀ ? [0, T].     

A conditional dichotomy theorem for stochastic processes with independent increments

Let X(t) and Y(t) be two stochastically continuous processes with independent increments over [0, T] and Lévy spectral measures M_t and N_t, respectively, and let the “time-jump” measures M and N be defined over [0, T] × $\text{R}$?{0} by M((t₁, t₂] × A) = M_t2(A) ? M_t1(A) and N((T₁, t₂] × A) = N_t2(A) ? N_t1(A). Under the assumption that M is equivalent to N, it is shown that the measures induced on function space by X(t) and Y(t) are either equivalent or orthogonal, and necessary and sufficient conditions for equivalence are given. As a corollary a complete characterization of the set of admissible translates of such processes is obtained: a function f is an admissible translate for X(t) if and only if it is an admissible translate for the Gaussian component of X(t). In particular, if X(t) has no Gaussian component, then every nontrivial translate of X(t) is orthogonal to it.     

When is c(x) a Clean Ring?

An element of a ring R is called clean if it is the sum of a unit and an idempotent and a subset A of R is called clean if every element of A is clean. A topological characterization of clean elements of C(X) is given and it is shown that C(X) is clean if and only if X is strongly zero-dimensional, if and only if there exists a clean prime ideal in C(X). We will also characterize topological spaces X for which the ideal C_K(X) is clean. Whenever X is locally compact, it is shown that C_K(X) is clean if and only if X is zero-dimensional.     

Some geometric conditions which imply the fixed point property for multivalued nonexpansive mappings

We show some geometric conditions on a Banach space X concerning the modulus of smoothness, the coefficient of weak orthogonality, the coefficient R(a,X), the James constant and the Jordan-von Neumann constant, which imply the existence of fixed points for multivalued nonexpansive mappings. These fixed point theorems improve some previous results and give affirmative answers to some open questions.     

Local selections and local dendrites

Let X be a Peano continuum, C(X) its space of subcontinua, and C(X, ε) the space of subcontinua of diameter less than ε. A selection on some subspace of C(X) is a continuous choice function; the selection σ is rigid if σ(A) ? B ? A implies σ(A) = σ(B). It is shown that X is a local dendrite (contains at most one simple closed curve) if and only if there exists ε > 0 such that C(X, ε) admits a selection (rigid selection). Further, C(X) admits a local selection at the subcontinuum A if and only if A has a neighborhood (relative to the space C(X)) which contains no cyclic local dendrite; moreover, that local selection may be chosen to be a constant.     

Realizing modules over the homology of a DGA

Let A be a DGA over a field and X a module over H_∗(A). Fix an A_∞-structure on H_∗(A) making it quasi-isomorphic to A. We construct an equivalence of categories between A_n+1-module structures on X and length n Postnikov systems in the derived category of A-modules based on the bar resolution of X. This implies that quasi-isomorphism classes of A_n-structures on X are in bijective correspondence with weak equivalence classes of rigidifications of the first n terms of the bar resolution of X to a complex of A-modules. The above equivalences of categories are compatible for different values of n. This implies that two obstruction theories for realizing X as the homology of an A-module coincide.     

Some properties concerning Milman's moduli

In this paper, some properties concerning Milman's moduli J(t,X) and S(t,X), including equivalent representations, upper and lower bounds as well as relationship between some geometric properties and these two moduli, are studied. In addition, some sufficient conditions for the fixed point property are presented.     

A Nagata-like theorem for certain function spaces

Let X be a space, and let A be a zero-dimensional topological ring. In this paper we will consider a few natural questions that arise when studying the space C _p (X, A), the ring of continuous functions from X to A, endowed with the topology of pointwise convergence. It will be shown that the zero-dimensionality of the codomain plays a vital role in this study. An upper and lower bound will be determined for the density of C _p (X, A) using the density of A and the weight of X. The character of C _p (X, A) will be computed, thus characterizing when C _p (X, A) is metrizable. Lastly, we will consider the topological dual space of C _p (X, A) and use it to prove a Nagata-like theorem.     

Left inverses of matrices with polynomial decay

It is known that the algebra of Schur operators on ?² (namely operators bounded on both ?¹ and ?^∞) is not inverse-closed. When ?²=?²(X) where X is a metric space, one can consider elements of the Schur algebra with certain decay at infinity. For instance if X has the doubling property, then Q. Sun has proved that the weighted Schur algebra Aω(X) for a strictly polynomial weight ω is inverse-closed. In this paper, we prove a sharp result on left-invertibility of the these operators. Namely, if an operator A∈Aω(X) satisfies ‖Afp_‖?‖fp_‖, for some 1?p?∞, then it admits a left-inverse in Aω(X). The main difficulty here is to obtain the above inequality in ?². The author was both motivated and inspired by a previous work of Aldroubi, Baskarov and Krishtal (2008) [1], where similar results were obtained through different methods for X=Zd, under additional conditions on the decay.     

On an additive representation function

Let A be an infinite subset of natural numbers, and X a positive real number. Let r(n) denotes the number of solution of the equation n=a₁+a₂ where a₁?a₂ and a₁, a₂∈A. Also let |A(X)| denotes the number of natural numbers which are less than or equal to X and belong to A. For those A which satisfy the condition that for all sufficiently large natural numbers n we have r(n)≠1, we improve the lower bound of |A(X)| given by Nicolas et. al. [NRS98]. The bound which we obtain is essentially best possible.     

Geometric Property （T）

This paper discusses“geometric property （T）”. This is a property of metric spaces introduced in earlier works of the authors for its applications to K-theory. Geometric property （T） is a strong form of “expansion property”, in particular, for a sequence （Xn） of bounded degree finite graphs, it is strictly stronger than （Xn） being an expander in the sense that the Cheeger constants h（Xn） are bounded below. In this paper, the authors show that geometric property （T） is a coarse invariant, i.e., it depends only on the large-scale geometry of a metric space X. The authors also discuss how geometric property （T） interacts with amenability, property （T） for groups, and coarse geometric notions of a-T-menability. In particular, it is shown that property （T） for a residually finite group is characterised by geometric property （T） for its finite quotients.     

Fixed points on model solvmanifold pairs

In this paper we define model solvmanifold pairs and their diagonal type selfmaps in the tradition of Heath and Keppelmann. We derive an explicit formula for computing the relative Nielsen number N(F;X,A) on these spaces and selfmaps F:(X,A)→(X,A). We find that model solvmanifold pairs often exhibit interesting Schirmer theory, meaning N(F;X,A)>max{N(F),N(F|_A)}.