The p-approximation property in terms of density of finite rank operators

We characterize the p-approximation property (p-AP) introduced by Sinha and Karn [D.P. Sinha, A.K. Karn, Compact operators whose adjoints factor through subspaces of ?p, Studia Math. 150 (2002) 17-33] in terms of density of finite rank operators in the spaces of p-compact and of adjoints of p-summable operators. As application, the p-AP of dual Banach spaces is characterized via density of finite rank operators in the space of quasi-p-nuclear operators. This relates the p-AP to Saphar's approximation property APp^′. As another application, the p-AP is characterized via a trace condition, allowing to define the trace functional on certain subspaces of the space of nuclear operators.     

Linear relations for p-harmonic functions

We discuss the p-harmonicity of the linear combination of p-harmonic functions in the Euclidean space and on a tree. If p≠2, the p-harmonicity is non-linear, i.e., the linear combination of p-harmonic functions need not be p-harmonic. In spite of this non-linear nature, we find some p-harmonic functions whose linear combinations become p-harmonic.     

On p-approximation properties for p-operator spaces

This paper has a two-fold purpose. Let 1<p<∞. We first introduce the p-operator space injective tensor product and study various properties related to this tensor product, including the p-operator space approximation property, for p-operator spaces on Lp-spaces. We then apply these properties to the study of the pseudofunction algebra PFp(G), the pseudomeasure algebra PMp(G), and the Figà-Talamanca-Herz algebra Ap(G) of a locally compact group G. We show that if G is a discrete group, then most of approximation properties for the reduced group C^∗-algebra , the group von Neumann algebra VN(G), and the Fourier algebra A(G) (related to amenability, weak amenability, and approximation property of G) have the natural p-analogues for PFp(G), PMp(G), and Ap(G), respectively. The p-completely bounded multiplier algebra McbAp(G) plays an important role in this work.     

On p-covalenced association schemes

Let (X,S) denote an association scheme where X is a finite set. For a prime p we say that (X,S) is p-covalenced (p-valenced) if every multiplicity (valency, respectively) of (X,S) is a power of p. In the character theory of finite groups Ito's theorem states that a finite group G has a normal abelian p-complement if and only if every character degree of G is a power of p. In this article we generalize Ito's theorem to p-valenced association schemes, i.e., a p-valenced association scheme (X,S) has a normal p-covalenced p-complement if and only if (X,S) is p-covalenced.     

Some bounds on the p-domination number in trees

Let p be a positive integer and G=(V,E) a graph. A subset S of V is a p-dominating set if every vertex of V-S is dominated at least p times, and S is a p-dependent set of G if the subgraph induced by the vertices of S has maximum degree at most p-1. The minimum cardinality of a p-dominating set a of G is the p-domination number γ_p(G) and the maximum cardinality of a p-dependent set of G is the p-dependence number β_p(G). For every positive integer p?2, we show that for a bipartite graph G, γ_p(G) is bounded above by (|V|+|Y_p|)/2, where Y_p is the set of vertices of G of degree at most p-1, and for every tree T, γ_p(T) is bounded below by β_p-1(T). Moreover, we characterize the trees achieving equality in each bound.     

Gaps of operators

We construct examples which distinguish clearly the classes of p-hyponormal operators for 0<p?∞. In addition, we show that those examples classify the classes of w-hyponormal, absolute-p-paranormal, and normaloid operators on the complex Hilbert space. Only a few examples of p-hyponormal operators have been examined. Our technique can provide many examples related to the above operators.     

Some paranormed sequence spaces of non-absolute type derived by weighted mean

The sequence spaces ?_∞(p), c(p) and c₀(p) were introduced and studied by Maddox [I.J. Maddox, Paranormed sequence spaces generated by infinite matrices, Proc. Cambridge Philos. Soc. 64 (1968) 335-340]. In the present paper, the sequence spaces λ(u,v;p) of non-absolute type which are derived by the generalized weighted mean are defined and proved that the spaces λ(u,v;p) and λ(p) are linearly isomorphic, where λ denotes the one of the sequence spaces ?_∞, c or c₀. Besides this, the β- and γ-duals of the spaces λ(u,v;p) are computed and the basis of the spaces c₀(u,v;p) and c(u,v;p) is constructed. Additionally, it is established that the sequence space c₀(u,v) has AD property and given the f-dual of the space c₀(u,v;p). Finally, the matrix mappings from the sequence spaces λ(u,v;p) to the sequence space μ and from the sequence space μ to the sequence spaces λ(u,v;p) are characterized.     

Density of finite rank operators in the Banach space of p-compact operators

A Banach space X is said to have the kp-approximation property (kp-AP) if for every Banach space Y, the space F(Y,X) of finite rank operators is dense in the space Kp(Y,X) of p-compact operators endowed with its natural ideal norm kp. In this paper we study this notion that has been previously treated by Sinha and Karn (2002) in [15]. As application, the kp-AP of dual Banach spaces is characterized via density of finite rank operators in the space of quasi p-nuclear operators for the p-summing norm. This allows to obtain a relation between the kp-AP and Saphar's approximation property. As another application, the kp-AP is characterized in terms of a trace condition. Finally, we relate the kp-AP to the (p,p)-approximation property introduced in Sinha and Karn (2002) [15] for subspaces of Lp(μ)-spaces.     

On some new generalized sequence spaces

In this paper we define the sequence sets ?_∞(u,Δ²,p), c(u,Δ²,p) and c₀(u,Δ²,p), and give α- and β-duals of these sets. Further we investigate matrix transformations in the spaces and give a characterization of the class (?_∞(u,Δ²,p),?_∞).     

p-adic Arakelov theory

We introduce the p-adic analogue of Arakelov intersection theory on arithmetic surfaces. The intersection pairing in an extension of the p-adic height pairing for divisors of degree 0 in the form described by Coleman and Gross. It also uses Coleman integration and is related to work of Colmez on p-adic Green functions. We introduce the p-adic version of a metrized line bundle and define the metric on the determinant of its cohomology in the style of Faltings. We also prove analogues of the Adjunction formula and the Riemann-Roch formula.     

The p/q-active uncapacitated facility location problem: Investigation of the solution space and an LP-fitting heuristic

The p/q-active uncapacitated facility location problem is the problem of locating p facilities on n possible sites each serving at least q of the m clients at the minimum cost. The problem is an extension of the uncapacitated facility location problem (UFL) where constraints on the number of facilities and their minimum activity have been added. A use of this formulation could be the opening of p new schools where each must have at least q pupils. p/q-active is NP-hard like the UFL.     

A class of B-(p,r)-invex functions and mathematical programming

Invexity of a function is generalized. The new class of nonconvex functions, called B-(p,r)-invex functions with respect to η and b, being introduced, includes many well-known classes of generalized invex functions as its subclasses. Some properties of the introduced class of B-(p,r)-invex functions with respect to η and b are studied. Further, mathematical programming problems involving B-(p,r)-invex functions with respect to η and b are considered. The equivalence between saddle points and optima, and different type duality theorems are established for this type of optimization problems.     

A note on p-adic q-ζ-functions II

We show that p-adic q-ζ-function constructed by Koblitz [7] (see also D?browski [4]) can be obtained as Γ-transform of some p-adic measure coming from Lubin–Tate formal group.     

Bounded linear non-absolutely summing operators

We show that, in certain situations, we have lineability in the set of bounded linear and non-absolutely summing operators. Examples on lineability of the set Πp(E,F)?Ip(E,F) are also presented and some open questions are proposed.     

Embeddings and frames of function spaces

For every Tychonoff space X we denote by Cp(X) the set of all continuous real-valued functions on X with the pointwise convergence topology, i.e., the topology of subspace of RX. A set P is a frame for the space Cp(X) if Cp(X)⊂P⊂RX. We prove that if Cp(X) embeds in a σ-compact space of countable tightness then X is countable. This shows that it is natural to study when Cp(X) has a frame of countable tightness with some compactness-like property. We prove, among other things, that if X is compact and the space Cp(X) has a Lindelöf frame of countable tightness then t(X)?ω. We give some generalizations of this result for the case of frames as well as for embeddings of Cp(X) in arbitrary spaces.     

On the behavior of some two-variable p-adic L-functions

We give some p-adic integral representations for the two-variable p-adic L-functions introduced recently by G. Fox. For powers of the Teichmüller character, we use the integral representation to extend the L-function to a larger domain, in which it is a meromorphic function in the first variable and an analytic element in the second. These integral representations imply systems of congruences for the generalized Bernoulli polynomials, improving previous results of Fox, Gunaratne, and the author; they also lead to generalizations of some formulas of Diamond and of Ferrero and Greenberg for p-adic L-functions in terms of the p-adic gamma and log gamma functions.     

Composition operators with weak hyponormality

There are many operator classes that are weaker than p-hyponormal. These include p-quasihyponormal, absolute p-paranormal, p-paranormal, normaloid, and spectraloid. In this note, we discuss measure theoretic composition operators in these classes.     

Some new spaces of double sequences

In this study, we define the double sequence spaces BS, BS(t), CSp, CSbp, CSr and BV, and also examine some properties of those sequence spaces. Furthermore, we show that these sequence spaces are complete paranormed or normed spaces under some certain conditions. We determine the α-duals of the spaces BS, BV, CSbp and the β(?)-duals of the spaces CSbp and CSr of double series. Finally, we give the conditions which characterize the class of four-dimensional matrix mappings defined on the spaces CSbp, CSr and CSp of double series.     

Modulability and duality of certain cones in pluripotential theory

Let p>0, and let Ep denote the cone of negative plurisubharmonic functions with finite pluricomplex p-energy. We prove that the vector space δEp=Ep−Ep, with the vector ordering induced by the cone Ep is σ-Dedekind complete, and equipped with a suitable quasi-norm it is a non-separable quasi-Banach space with a decomposition property with control of the quasi-norm. Furthermore, we explicitly characterize its topological dual. The cone Ep in the quasi-normed space δEp is closed, generating, and has empty interior.     

On the T-ramified, S-split p-class field towers over an extension of degree prime to p

Let K be a number field, p a prime, and let be the T-ramified, S-split p-class field tower of K, i.e., the maximal pro-p-extension of K unramified outside T and totally split on S, where T and S are disjoint finite sets of places of K. Using a theorem of Tate on nilpotent quotient groups, we give (Theorem 2 in Section 3) an elementary characterisation of the finite extensions L/K, with a normal closure of degree prime to p, such that the analogous p-class field tower of L is equal to the compositum . This N.S.C. only depends on classes and units of L. Some applications and examples are given.