Summability and estimates for polynomials and multilinear mappings

In this paper we extend and generalize several known estimates for homogeneous polynomials and multilinear mappings on Banach spaces. Applying the theory of absolutely summing nonlinear mappings, we prove that estimates which are known for mappings on ?_p spaces in fact hold true for mappings on arbitrary Banach spaces.     

Dominated, diagonal polynomials on ℓ p spaces

We show that the r-dominated polynomials on _p(2 p ) are integral on ₁, and give examples proving that the converse is not true. We characterize when the 2-homogeneous, diagonal polynomials on _p(1 < p ) are r-dominated. We prove that, unlike the linear case, there are nuclear polynomials which are not 1-dominated.Received: 6 June 2004; revised: 28 September 2004     

On ideals of polynomials and multilinear mappings between Banach spaces

It is shown that for every quasi-normed ideal ${\cal Q}$ of n-homogeneous continuous polynomials between Banach spaces there is a quasi-normed ideal ${\cal A}$ of n-linear continuous mappings ${\cal A}$ such that $q \in {\cal Q}$ if and only if the associated n-linear mapping $\check{q}$ of q is in ${\cal A}$. Received: 12 March 2001     

Polynomial numerical indices of Banach spaces with absolute norm

We study when a Banach space with absolute norm may have polynomial numerical indices equal to one. In the real case, we show that a Banach space X with absolute norm, which has the Radon-Nikodým property or is Asplund, satisfies n⁽²⁾(X)<1 unless it is one-dimensional. In the complex case, we show that the only Banach spaces X with absolute norm and the Radon-Nikodým property which satisfy n⁽²⁾(X)=1 are the spaces . Also, the only Asplund complex space X with absolute norm which satisfies n⁽²⁾(X)=1 is c₀(Λ).     

Two new properties of ideals of polynomials and applications

In this paper we introduce two properties for ideals of polynomials between Banach spaces and showhow useful they are to deal with several a priori different problems. By investigating these properties we obtain, among other results, new polynomial characterizations of L_∞ spaces and characterizations of Banach spaces whose duals are isomorphic to f 1 (Λ).     

Factorization of entire mappings of nuclear bounded type

In this paper we study the relationships between the spaces of entire mappings of bounded type, entire mappings of nuclear bounded type, entire mappings of Pietsch integral bounded type, and entire mappings of Grothendieck integral bounded type. Several results due to Alencar (Proc. Roy. Irish Acad.85A (1985) 131-138) and Cilia and Gutiérrez (J. Aust. Math. Soc.76 (2004) 269-280) for homogeneous polynomials are extended to entire mappings. In the main result we prove that an entire mapping is of nuclear bounded type if and only if it factors through an entire mapping of Pietsch integral bounded type.     

Dominated polynomials on $\mathcal{L}_P $ -spaces

It is well known that continuous bilinear forms on C(K) × C(K) are 2-dominated. This paper shows that generalizations of this result are not to be expected. The main result asserts that for every -space E(1 p ), every n 2, every r > 0 and every Banach space F , there exists an n-homogeneous polynomial P : E F such that P is not of type [_r], hence P is neither r-dominated nor r-semi-integral (if n = 2 and p = , F is supposed to contain an isomorphic copy of some , 1_q < ).Received: 24 November 2003     

Factoring operators over Hilbert-Schmidt maps and vector measures

We study the structure of Banach spaces X determined by the coincidence of nuclear maps on X with certain operator ideals involving absolutely summing maps and their relatives. With the emphasis mainly on Hilbert-space valued mappings, it is shown that the class of Hilbert—Schmidt spaces arises as a ‘solution set’ of the equation involving nuclear maps and the ideal of operators factoring through Hilbert—Schmidt maps. Among other results of this type, it is also shown that Hilbert spaces can be characterised by the equality of this latter ideal with the ideal of 2-summing maps. We shall also make use of this occasion to give an alternative proof of a famous theorem of Grothendieck using some well-known results from vector measure theory.     

Extension of Locally Defined Indefinite Functions on Ordered Groups

We give a definition of -indefinite function of archimedean type, on an interval of an ordered group with an archimedean point. We say that has the indefinite extension property if every continuous -indefinite function of archimedean type, on an interval of , can be extended to a continuous -indefinite function on the whole group .We show that if a group is semi-archimedean and it has the indefinite extension property, then with the lexicographic order and the product topology has the indefinite extension property. As a corollary it is obtained that the groups and with the lexicographic order and the usual topologies, have the indefinite extension property.To Professor José R. León, for his kind encouragement.Both authors were supported in part by the CDCH of the Univ. Central de Venezuela and by CONICIT grant G-97000668. Both authors were visitors at IVIC during the realization of this paper.Submitted: August 8, 2002 Revised: January 30, 2003     

Characterizations of new Cohen summing bilinear operators

We prove two characterizations of new Cohen summing bilinear operators. The first one is: Let X, Y and Z be Banach spaces, 1 < p < ∞, V : X × Y → Z a bounded linear operator and n ≥ 2 a natural number. Then V is new Cohen p-summing if and only if for all Banach spaces X₁,?…?, X_n and all p-summing operators U : X₁ × · · · × X_n → X, the operator V ? (U, I_Y) : X₁ × · · · × X_n × Y → Z is -summing. The second result is: Let H be a Hilbert space,, Y, Z Banach spaces and V : H × Y → Z a bounded bilinear operator and 1 < p < ∞. Then V is new Cohen p-summing if and only if for all Banach spaces E and all p-summing operators U : E → H, the operator V ? (U, I_Y) is (p, p*)-dominated.     

<Emphasis Type="Italic">R</Emphasis>-bounded Representations of <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript> (<Emphasis Type="Italic">G</Emphasis>)

We investigate R-bounded representations , where X is a Banach space and G is a lca group. Observing that Ψ induces a (strongly continuous) group homomorphism , we are then able to analyze certain classical homomorphisms U (e.g. translations in L^p (G)) from the viewpoint of R-boundedness and the theory of scalar-type spectral operators. Dedicated to the memory of H. H. Schaefer     

The Ptolemy and Zb?ganu constants of normed spaces

In every inner product space H the Ptolemy inequality holds: the product of the diagonals of a quadrilateral is less than or equal to the sum of the products of the opposite sides. In other words, ‖x−y‖‖z−w‖≤‖x−z‖‖y−w‖+‖z−y‖‖x−w‖ for any points w,x,y,z in H. It is known that for each normed space (X,‖⋅‖), there exists a constant C such that for any w,x,y,z∈X, we have ‖x−y‖‖z−w‖≤C(‖x−z‖‖y−w‖+‖z−y‖‖x−w‖). The smallest such C is called the Ptolemy constant of X and is denoted by C_P(X). We study the relationships between this constant and the geometry of the space X, and hence with metric fixed point theory. In particular, we relate the Ptolemy constant C_P to the Zb?ganu constant C_Z, and prove that if X is a Banach space with , then X has (uniform) normal structure and therefore the fixed point property for nonexpansive mappings. We derive general lower and upper bounds for both C_P and C_Z, and calculate the precise values of these two constants for several normed spaces. We also present a number of conjectures and open problems.     

Image of the Spectral Measure of a Jacobi Field and the Corresponding Operators

By definition, a Jacobi field is a family of commuting selfadjoint three-diagonal operators in the Fock space The operators J(ϕ) are indexed by the vectors of a real Hilbert space H₊. The spectral measure ρ of the field J is defined on the space H₋ of functionals over H₊. The image of the measure ρ under a mapping is a probability measure ρ_K on T₋. We obtain a family J_K of operators whose spectral measure is equal to ρ_K. We also obtain the chaotic decomposition for the space L²(T₋, d_{ρ K}).     

Universal mappings for certain classes of operators and polynomials between Banach spaces

A well‐known result of J. Lindenstrauss and A. Pełczyński (1968) gives the existence of a universal non‐weakly compact operator between Banach spaces. We show the existence of universal non‐Rosenthal, non‐limited, and non‐Grothendieck operators. We also prove that there does not exist a universal non‐Dunford–Pettis operator, but there is a universal class of non‐Dunford–Pettis operators. Moreover, we show that, for several classes of polynomials between Banach spaces, including the non‐weakly compact polynomials, there does not exist a universal polynomial.     

Orthogonally Additive Polynomials over C(K) are Measures—A Short Proof

We give a simple proof of the fact that orthogonally additive polynomials on C(K) are represented by regular Borel measures over K. We also prove that the Aron-Berner extension preserves this class of polynomials.     

Almost summing mappings

We introduce a general definition of almost p-summing mappings and give several concrete examples of such mappings. Some known results are considerably generalized and we present various situations in which the space of almost p-summing multilinear mappings coincides with the whole space of continuous multilinear mappings. Received: 17 June 2002     

Spectral measures,III: Densely defined spectral measures

In this paper we extend the theory of spectral measures developed in Parts I and II to the case where values are assumed in the set of discontinuous (in normed spaces „unbounded”) operators. Examples of operators in nonlocally convex spaces are given, which have densely defined measures.     

Absolutely γ-summing multilinear operators

In this paper we introduce an abstract approach to the notion of absolutely summing multilinear operators. We show that several previous results on different contexts (absolutely summability, almost summability, Cohen summability) are particular cases of our general results.